Science Department - ISP

Chapter 1 - Review of Introductory Concepts and Mathematical Tools Used in Physics

1.1 - Introductory Concepts

Physics: Science that deals with the relationships between matter, energy, space, and time, and their structure. It is the study and analysis of all the natural phenomena that have mathematical structure.

Energy: The ability to cause changes in matter.

Matter: Everything, which occupies space, can be perceived by one or more senses (directly or indirectly), and constitutes a physical body; it is everything that has the attributes of inertia and interaction.

Inertia: Tendency of a piece of matter to resist to changes in its state of motion.

Interaction: Ability of matter to "detect" the presence or existence of other pieces of matter.

Basic Interactions in Nature:

1.- Gravitational: Interaction between two objects that occurs because they both have mass. The gravitational interaction is due exclusively to the existence of mass, is always attractive, operates over very long distances, and is the weakest of all the four basic interactions observed in nature.

2.- Electromagnetic: Ability of an object to interact with another object because they are both electrically charged. The electromagnetic interaction is due exclusively to the existence of the electric charge. Since two types of electric charge have been observed in nature, it manifests in two different ways: attraction or repulsion. It is about 10E40 times greater than the gravitational interaction; it operates over all distances, but falls off with the square of the distance between the objects.

3.- Strong: Interaction that holds the particles in the atomic nucleus together. It occurs between a type of elementary particles called hadrons (like the proton, the neutron, and the pion), and is about 100 times greater than the electromagnetic interaction. It operates at very short range (about 10E–15 m).

4.- Weak: Interaction that occurs between a type of elementary particles called leptons (like the electron, the muon, and the neutrino), that is about 10E10 times weaker than the electromagnetic interaction.

Time: The continuous, forward flow of events.

Space: A limited extension that may extend in one, two, or three dimensions; it extends with no bounds in all directions and is the field of physical objects, events, and their order and relationship.

1.2 - Mathematical Tools

Unit: Value or quantity in terms of which other values or quantities are expressed. A unit is fixed by definition.

Significant Figures: Digits that are known with certainty plus the first uncertain digit in a measurement. In an experimental measurement, they are all the numbers that can be read directly from the instrument's scale plus one doubtful or estimated digit.

1.3.1 - Error Theory

Positive Error: Errors that tend to make observations always too high.

Negative Errors: Errors that tend to make observations always too low.

Systematic Errors: Errors that tend to make all observations made always positive or always negative. Their causes are known, so they may be estimated and corrected. They are always associated with a particular instrument or technique. Avoiding systematic errors depends on the skills of the observer to detect, prevent and correct them. They are usually grouped in 3 different categories: instrumental, personal, and external.

(1) Instrumental: Produced by faults in the apparatus used. To reduce or correct them, the instruments should be checked with accurate standards and the necessary corrections applied to the observations.

(2) Personal: Depend upon the particular way in which the observer takes the data. They may be minimized by taking observations under various conditions and by using several observers working independently.

(3) External: Usually caused by conditions over which the observer has no control, but that can be estimated. Therefore, they cannot be eliminated, but the necessary corrections may be applied.

Random Errors: Errors that result from unknown and unpredictable variations in experimental situations, sometimes beyond the control of the observer. Usually due to a large number of factors, each of which adds its own small contribution to the total error. Positive and negative errors are equally probable, so random errors are subject to the laws of chance. Random errors can be minimized by improving and refining experimental techniques and by repeating the measurement a large number of times, so erroneous readings become statistically insignificant.

Accuracy: Measures the correctness of an experimental result; this is, how close the experimental result comes to the true value. The accuracy of an experimental value depends on systematic errors. When a measurement has no systematic error we say it is exact.

Precision: Measures the reliability of an experimental result; this is, it measures the magnitude of the uncertainty of the result (how close different outcomes are). When random errors are minimized, we say that the result is precise.

1.3.2 - Statistical Quantification of Errors

Most Probable Value: Also known as average value. For a list of N experimental measurements, it is obtained by adding all the measurements and dividing the result by the quantity of measurements that were added.

Deviation: Quantity that tells how much a particular measurement scatters from the mean. A deviation may be positive or negative, so the sum of the deviations is expected to be (ideally) 0.

Average (or Mean) Deviation: Quantity that measures the dispersion of a set of experimental measurements from the mean (a measure of precision).

Variance: Average of the squares of the deviations of a set of measurements; it is used to avoid the problem of negative deviations and absolute values.

Standard Deviation: Quantity that gives the absolute error of a set of measurements. It describes the precision of the mean of a set of measurements. Operationally defined as the square root of the variance. For a small number of measurements, it can be statistically shown that a better value for the standard deviation is obtained if we divide the su of the squares of the deviations by N-1 instead of N.

Accepted or "True" Value: Most accurate value of a quantity that is usually obtained through sophisticated experiments or mathematical methods. This is the value found on textbooks and handbooks.

Absolute Difference: Absolute value of the difference between an experimental value and the accepted value of a quantity.

Fractional Error: Ratio of the absolute difference to the accepted value.

Percent Error: Most common way of expressing the fractional error, given by the fractional error X 100%

Percent Difference: Ratio of the absolute difference between two experimental values to their average. When there are more than two measurements, the PD is found by dividing the absolute value of the difference of the extreme values by the average of all the measurements.

1.5 - Vector Analysis

Vector Analysis: Mathematical formalism that tells us how to use mathematical objects called vector, which have direct application in physics and engineering.

Scalar: Quantity that has only magnitude; it requires only a number to completely define it.

Vector: Quantity that has two characteristics: magnitude and direction.

Vector Algebra: Mathematical theory developed for vectors. It is the group of rules that can be applied to the set of mathematical objects called vectors.

Scalar Product: Process for multiplying two vectors in such a way that a scalar is obtained. Operationally, it is defined as the product of the magnitude of the two vectors and the cosine of the angle between them.

Vector Product: Process for multiplying two vectors in such a way that they produce a new vector. The magnitude of the new vector is the product of the magnitudes of the two vectors multiplied and the sine of the angle between them. The direction of the resulting vector is perpendicular to the plane formed by the vectors that were multiplied, as given by the right-hand-rule. Operationally,

Chapter 2 - Kinematics I

Mechanics: Study of motion, forces, and their effects on objects.

Kinematics: Study of how things move without considering the causes of this motion.

Dynamics: Study of why things change their state of motion; study of forces and the changes in motion that they cause.

Statics: Particular case of dynamics that deals with the situations in which the forces acting on an object or system produce no change in its state of motion (equilibrium).

Distance: The length of the path between two points.

Displacement: Vector form of the distance between two points; it is the change of position in a particular direction.

Velocity: The rate of change of position; it is a vector that gives the displacement per unit time.

Speed: Magnitude of the velocity vector. Given by v = d/t, where d is the distance traveled by the object, measured from the starting point to the final point of the motion, and t is the time taken to go from the starting point to the final point.

Average Speed: Ratio of the total distance traveled x to the time interval t. It is the constant speed that represents the steady motion of an object during an interval of time.

Instantaneous Velocity: Velocity at one instant in time.

Acceleration: Time rate at which velocity changes. When considering motion in one dimension, it is the rate of increase of speed. As a scalar (pick-up), it may be positive or negative. Negative values mean that the speed is decreasing and may be called deceleration or retardation. As a vector quantity, whenever speed changes, or direction changes, or both speed and direction change, there is an acceleration.

Chapter 3 - Kinematics II

Frames of Reference: Any system for specifying the precise location of objects in space. Ordinarily, set of coordinate axes is used.

Projectile Motion: Type of motion in which objects are given an initial velocity and then move only under the force of gravity, describing parabolic paths.

Chapter 4 - Dynamics

Dynamics: Study of why things change their state of motion; study of forces and the charges in motion that they cause.

Force: Interaction between two bodies. It is the physical quantity that can affect the motion of an object. Forces are the causes of the effects we call accelerations. They are vector quantities whose magnitude is measured by the magnitude of the acceleration they can impart to a given body; the direction of a force is parallel to the acceleration it gives to an object.

Newton's Laws of Motion

(1) First Law of Motion - Inertia: "A body remains at rest, or if in motion, it remains in uniform motion with constant speed in a straight path, unless it is acted on by an unbalanced external force."

(2) Second Law of Motion - Causality: "The acceleration produced by an unbalanced force acting on a body is proportional to the magnitude of the net force, in the same direction as the force, and inversely proportional to the mass of the body."

(3) Third Law of Motion - Interaction: "Whenever one body exerts a force on a second body, the second body exerts a force on the first body; these forces are equal in magnitude, but opposite in direction."

Weight: The force with which a mass is attracted by the Earth, given by w = mg.

Normal Force: Force that is perpendicular to the surface producing it; it arises when two objects are in contact.

Free body Diagram: Diagram that shows all the forces acting in separate parts of the system.

Friction: Force that opposes to motion when one body slides over another. Frictional forces are always parallel to the surfaces that are in contact.

Static Friction: Frictional force between two bodies at rest; it is directly proportional to the normal force. It is the maximum friction that acts between the objects.

Kinetic Friction: Frictional force between two bodies when one slides on the other. It is constant and independent of the relative velocity of the surfaces.

Chapter 5 - Circular Motion and Gravitation

Uniform Circular Motion: The form of motion of an object moving at constant speed v in a circle of radius r. For this to be possible, a force must act on the object.

Period (T): Time used to make one complete revolution.

Newton's Law of Universal Gravitation

"The gravitational force of attraction between any two objects in the universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers."

Kepler's Laws of Planetary Motion

First Law - Law of Orbits: The planets move in elliptical orbits with the sun at one focus.

Second Law - Law of Areas: A line (or position vector) joining any planet to the sun sweeps out equal areas in equal intervals of time.

Third Law - Law of Periods: The square of the period of revolution of any planet is proportional to the cube of the mean distance of the planet from the sun.

Chapter 6 - Work and Energy

Work: Scalar quantity given by the product of the magnitude of the displacement of an object and the component of the force parallel to that displacement.

Energy: Ability of a body to do work.

Kinetic Energy: Energy a body has because of its motion. If a body of mass m is moving at a speed v, its translational kinetic energy is ½ mv2.

Potential Energy: Energy associated with the position or configuration of a system of bodies. Since it is the energy of a system of bodies due to the relative positions of the parts of the system, at least two bodies are needed to define the PE.

Law of Conservation of Energy

"The total energy is neither increased nor decreased in any process. Energy can be transformed from one form into another, and transferred from one body to another, but the total amount of mechanical, thermal chemical, electrical, and other forms of energy in an isolated system remains constant."

Power: Rate at which work is done, or rate at which energy is transformed.

Chapter 7 - Momentum

Linear Momentum: Product of the mass of an object and its velocity.

Newton's Second Law of Motion in Terms of Momentum

"The rate of change of momentum of a body is proportional to the net force applied to it."

Law of Conservation of Linear Momentum

"The total linear momentum of an isolated system of bodies remains constant."

System: Set of objects that may interact with each other.

Isolated System: System in which the only forces present are those between the objects of the system; no external force acts on them.

Impulse: Change in the momentum of an object, produced by a force acting during a short interval of time.

Elastic Impact: Collision in which the total kinetic energy is conserved; this is, a collision in which no KE is transformed to a non-mechanical form of energy.

Inelastic Impact: Collision in which some KE is transformed into other forms of energy that are not mechanical; this means that, from the mechanical perspective, the KE is not conserved.

Coefficient of Restitution: Quantity that measures the elasticity or inelasticity of a collision; it depends on the nature of the colliding bodies.

Center of Mass: Point in a body or a system at which the whole mass can be considered to act. It coincides with the center of symmetry if the body has a uniform density throughout.

Center of Gravity: Point in a body at which the force of gravity can be considered to act. It can be assumed that the entire weight of the body acts at this point.

Chapter 8 - Rotational Motion

Rotational Motion: Motion of a body turning about an axis; all points in the body move in circles such that the centers of these circles all lie on the axis of rotation.

Rigid Body: Body with a definite shape that does not change, so that the particles composing it stay in fixed positions relative to one another.

Angular Velocity: Rate of change of angular displacement.

Angular Acceleration: Rate of change of angular velocity.

Frequency: Number of complete revolutions per second.

Torque: Also known as moment due to a force about an axis. It is a measure of the effectiveness of a force in producing rotation about a given axis. Operationally, it is defied as the product of the force and the perpendicular distance from the axis of rotation and the line of action of the force. This perpendicular distance is called the lever arm or torque arm.

Rotational Inertia: Tendency of a body to resist a change in its angular velocity (state of rotation). It depends upon the axis of rotation and the way in which the masses are distributed relative to it.

Angular Momentum: Product of the moment of inertia of a body and its angular velocity.

Law of Conservation of Angular Momentum

"If the net external torque acting on a system about a given axis is zero, the angular momentum about that axis will be constant."

Chapter 9 - Statics and Elasticity

Statics: Particular case of dynamics that deals with the situations in which the forces acting on an object or system produce no change in its state of motion; special case of dynamics in which the acceleration is 0.

Equilibrium: Condition of a body whose velocity is constant in magnitude and direction. This includes the case in which the velocity is zero and remains so. State of a body in which there is no change in its motion.

Simple machine: Any device that transmits the application of a force into useful work; it multiplies force at the expense of distance or multiplies distance at the expense of force.

Efficiency: Ratio of the useful work output of a machine to the total work input.

Actual Mechanical Advantage (AMA): Ratio of the output force to the input force.

Ideal Mechanical Advantage (IMA): Ratio of the distance the input force moves to the distance the output force moves.

Lever: Rigid bar pivoted at a fulcrum. The force ratio and the distance ratio depend on the relative position of the pivot, the point where the user exerts the effort, and the point where the lever applies force to the load.

Pulley: Simple machine in which the power is transferred through the tension in a string wound over one or more wheels. They may be fixed or moving.

Gear: Notched wheel that can transmit torque by meshing with another notched wheel.

Incline Plane: Simple machine that can be used to raise a weight vertically by movement up an incline. Both distance ratio and force ratio depend on the angle of inclination.

Equilibrium: State of a body in which there is no change in its motion.

Stability: Measure of how hard it is to displace an object or system form equilibrium.

Stable Equilibrium: Equilibrium such that if the system is disturbed a little, there is a tendency for it to return to its original state.

Unstable Equilibrium: Equilibrium such that if the system is disturbed a little, there is a tendency for it to move further from its original position rather than to return.

Neutral Equilibrium: Equilibrium such that if the system is disturbed a little, there is no tendency for it neither to move further nor to return.

Ultimate strength: Maximum force that can be applied to a material without breaking it.

Stress: Distorting force per unit area.

Strain: Fractional change in dimensions produced by a stress applied to a body.

Elastic (Young’s) Modulus: Ratio of stress to strain in a solid; property characteristic of a given material that is independent of the object’s size or shape.

Tensile stress: Stress that stretches a body, which occurs when internal forces act inward to the body.

Compressive stress: Stress that compresses a body, which occurs when internal forces act inward to the body.

Shear stress: Stress that causes the deformation of a body without any change in volume. It occurs when opposite forces are applied across opposite faces of the object.

Shear strain: Ratio of the amount of deformation of the side of the body to the length of the side.

Shear Modulus: Constant of proportionality characteristic of a given material that measures the ratio of the tangential force per unit area to the angular deformation of the object.

Bulk Modulus: Constant of proportionality characteristic of a material given by the ratio of the change in the pressure acting on an object to the fractional change in volume it produced.

Pressure: Force per unit area.

Bulk Modulus: Constant of proportionality characteristic of a material, given by the ration of the change in the pressure acting on an object to the fractional change in volume it produces.

Chapter 10 - Fluid Mechanics

Phase (state): A condition in which we find matter.

Solid: Have fixed shape and volume, and they are very difficult to compress; atoms or molecules occupy fixed positions in space.

Liquid: Have fixed volume but does not maintain a fixed shape; they take the shape of its container. Not easily compressed, and its volume can only be changed by very large forces. The atoms and molecules move about at random, but they are quite close to one another and the motion is hindered.

Gas: Have no fixed shape or volume; they expand spontaneously to fill the container and are easily compressed. The molecules have almost free random motion.

Plasma: It is essentially a very hot gas that is capable of conducting an electric current, made of a mixture of free electrons and ions or atomic nuclei.

Fluid: Collective name given to liquids and gases, since they don’t have the ability to maintain a fixed length or shape because atoms and molecules move around in a relatively free way due to low cohesive forces.

Density: Characteristic property of a pure substance at a given temperature, given by its mass per unit volume.

Relative Density: Also called Specific Gravity. Ratio of the density of a substance to the density of water at 4°C, which is accepted to be 1.00 g/cm3 (1.00 x 103 kg/m3).

Pressure: Ratio of force per unit area when force is acting perpendicular to the surface area.

Incompressible fluid: Fluid whose density is constant and does not change with depth.

Pascal’s Principle: “The pressure applied to a confined fluid increases the pressure throughout by the same amount”.

Buoyancy: Tendency of a body to float or rise when submerged in a fluid.

Buoyant force: Upward force exerted by a fluid on an object. It occurs because the pressure in a fluid increases with depth, so the upward pressure in the bottom surface of an object in a fluid will be greater than the downward pressure on its top surface.

Archimedes’ Principle: “The buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by that object”.

Hydrostatics: Study of fluids in equilibrium.

Hydrodynamics: Study of fluids in motion.

Laminar or Streamline Flow: Smooth and steady flow in which neighboring layers of fluid slide by each other easily; each particle of the fluid follows a smooth path (streamline), and the paths do not cross each other.

Turbulent Flow: Fluid flow in which the speed at any point varies rapidly in magnitude and direction. It is characterized by whirlpool-like circles called eddy currents.

Viscosity: Resistance of a fluid to flow at low speeds. It exists in both liquids and gases, and is essentially a frictional force between different layers of fluid as they move past another. In liquids, it is due to the forces between molecules; in gases, it arises form collisions between the molecules.

Mass flow rate: Mass of fluid that passes a given point per unit time.

Volume rate of flow: Volume of fluid passing a given point per unit time.

Bernoulli’s Principle: “Where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high”.

Chapter 11 - Vibrations and waves.

Vibration (or oscillation): Any regularly repeated back and forth motion or change that occurs over the same path.

Periodic motion: Motion repeated in each of a succession of equal intervals of time.

Equilibrium position: Midpoint of the path of an object describing periodic motion.

Restoring force: force that tries to return an object to its equilibrium position.

Terms Used to Discuss Vibrational Motion

1.- Amplitude: Maximum displacement from the equilibrium position

2.- Cycle: Complete to-and-fro motion from some point back to the same initial point.

3.- Period: Time required for on complete cycle.

4.- Frequency: Number of complete cycles per second.

Simple Harmonic Motion: Motion of a vibrating system for which the acceleration is directly proportional to the displacement from an equilibrium position and is directed towards that position.

Simple Pendulum: Small mass oscillating back and forth at the end of a very light string. It the amplitude of the oscillation is small enough, it moves in approximately SHM.

Damped oscillation: Oscillation with an amplitude that progressively decreases in time, due to air resistance and internal friction.

Overdamped: If the damping is so large that it takes a long time to reach equilibrium.

Underdamped: If the system makes several oscillations before coming to rest.

Critically damped: If the motion comes to zero in the shortest possible time without completing the vibration.

Natural Frequency: Frequency at which an object or system will vibrate freely. This occurs when no external periodic force acts on the system and there is little air resistance.

Forced Vibration: Oscillation of an object or system subjected to an external periodic force. The oscillations have the same frequency as the applied force, and the amplitude depends upon the difference between the driving frequency and the natural frequency of the system.

Resonance: The largest-amplitude vibration of an object or system when given impulses at its natural frequency.

Waves: Oscillations that propagate without carrying matter with them, transferring energy from one place to another.

Wave motion: Vibrational motion of waves by which disturbance of equilibrium is produced; all forms of wave motion transport energy.

Pulse: Single, non-repeated disturbance.

Continuous or periodic wave: Disturbance that repeats on each of a succession of equal intervals of time.

Important Quantities

1.- Amplitude: Maximum height of a crest relative of the normal, undisturbed level.

2.- Wavelength: Distance between two successive crests, or any two successive points moving equally on the wave.

3.- Frequency: Number of complete waves that pass a given point per unit time.

4.- Period: Time elapsed between two successive crests passing by the same point in space.

5.- Wave velocity: Velocity at which a given part of the wave propagates through a medium.

Types of Waves

1.- Transverse waves: Waves in which vibrations of the medium are at right angles to the direction of propagation of the waves.

2.- Longitudinal waves: Waves in which the vibrations of the medium are parallel to the direction of propagation of the waves.

Surface waves: Waves that travel along the boundary between two materials.

Intensity of a wave: Power transmitted by a wave across a unit area perpendicular to the direction of the energy flow.

Reflection: Process in which a wave that reaches the boundary between two media “bounces back” to stay in the first medium.

Phase: Term used to describe the relative position of analog points of two different waves. It is the stage in a cycle that a wave has reached at a particular time.

Wave front: Continuous surface associated with a wave in which all the vibrations concerned are in phase.

Ray: Line drawn in the direction of motion of a wave, perpendicular to its wave front.

Angle of incidence: Angle that the incident ray makes with the line perpendicular to the surface defined by the boundary.

Refraction: Change in direction experienced by a wave disturbance as it passes obliquely from on medium into another in which the disturbance has a different velocity.

Interference: The superposing of one wave on another that occurs when two waves pass through the same region of space at the same time. The resultant displacement at any point during overlapping is the sum of the individual displacements.

Destructive interference: When the amplitude of a wave interfering is opposite to the amplitude of the other; the resultant displacement is less than the individual pulses.

Constructive interference: When the amplitudes of both interfering waves are in the same direction; he resultant displacement is greater than the individual pulses.

Superposition principal: When two waves interfere, the resultant displacement at a given point is the algebraic sum of the separate displacements of the individual waves at that point.

Diffraction: The spreading of a wave disturbance into a region behind an obstacle.

Standing waves: The interference effect resulting from two waves of the same type and equal frequency and intensity, but moving in opposite direction in the same region. The effect is most often caused when a wave is reflected back along its original path.

Fundamental frequency: Lowest frequency corresponding to one single antinode.

Important to remember: The energy transported by a wave is proportional to the square of the amplitude.

Laws of reflection:

1.-The angle of incidence equals the angle of reflection.

2.-The incident ray, the reflected ray, and the normal, all lie on the same plane.

Chapter 24 - The wave nature of light

Wave Front: Locus of points of a wave having the same phase of vibration.

Huygens’ Principle: “Every point on a wave front can be considered as a source of tiny wavelets that spread out in the forward direction at the speed of the wave itself so the new wave front is the envelope of the wavelets (tangent to all of them).”

Isotopic Medium: Medium in which waves travel equally fast in all directions.

Index of refraction: Measure of the optical density of a substance.

Snell’s Law: “The product of the index of refraction (for a given frequency of light) and the sine of the angle from the normal is constant, the same in each medium.”

Diffraction: Effect observed when a wave disturbance spreads into a region behind an obstacle. Its is due to the interference between the component ray of a single broad wave front.

Polarization: A restriction in the vibrations in a transverse wave. Normally in a transverse wave, the vibrations can have any direction in the plane perpendicular to the direction of propagation. If the direction is restricted in any way, the wave is said to be polarized. In a plane/polarized wave, the oscillations are restricted to a given plane.

Chapter 12 - Sound and its Characteristics

Sound: series of disturbances in matter that stimulate the human ear.

3 aspects are required for a sound to occur:

1.- Source: We need to have a sourde that creates the sound. Sound sources are vibrating objects.

2.- Waves: The source should be able to produce longitudinal waves, since this is the way in which energy is transferred from the source.

3.- Detector: A detector, like the human ear or any other instrument, is needed to detect the sound.

Pitch: The identification of a certain sound with a definite tone; it depends on the frequency that the ear receives. The pitch refers to high or how low the sound is.

Audible Range: Range of frequencies to which the human ear responds, typically between 20 Hz and 20 kHz.

Infrasonic Range: Vibrations in matter whose frequency is below the audible range (less than 20 Hz).

Ultrasonic Range: Vibrations in matter that occur at frequencies above 20 kHz.

Supersonic: Motion that takes place at speeds faster than that of sound.

* Do not confuse the terms ultrasonic and supersonic. Ultrasonic refers to frequency; supersonic refers to speed.

Intensity: Power transferred per unit area perpendicular to the direction of the energy flow.

Loudness: Sensation that depends principally on the intensity of sound waves reaching the ear.

Pressure Amplitude: Pressure difference between normal atmospheric pressure and the pressure peak produced by a sound wave.

Important Facts About Open Pipes

* They have displacement antinodes at both ends.

* At least one node must be present to produce a standing wave.

* Since the distance between two successive antinodes (or nodes) is (wavelength)/2, then L = (wavelegth)/2 and the fundamental frequency is f1 = v/(wavelength 1) = v/2L.

* For the first over tone, (wavelength 2) = L and f2 = v/L.

* The frequency of each overtone is an integer multiple of the fundamental frequency, so L = n (wavelength n)/2 or (wavelength n) = 2L/n, n = 1, 2, 3, . . .

Important Facts About Closed Pipes

* There is always a node at the closed end and an antinode at the open end.

* Since the smallest node-antinode distance is (wavelength)/4, L = (wavelength)/4 and the fundamental frequency in a closed pipe is f1 = v/(wavelength 1) = v/4L. This is half of what we get from an open tube of equal length.

* Only odd harmonics are present in a closed pipe. Overtones have frequencies equal to 3, 5, 7, . . . times the fundamental: fn = n(v/4L), n = 1, 3, 5, 7, . . .

* There is no way for waves with frequencies 2, 4, 6, . . . times the fundamental to have a node at one end and an antinode at the other end, so they cannot exist as standing waves in a closed tube.

Quality of Sound: Characteristic of a musical note that is determined by the frequencies present. It enables the listener to tell the difference between two notes of the same fundamental frequency played on different musical instruments.

Noise: Sound composed of a random mix of frequencies. It may affect us both psychologically and physiologically.

Interference: When two similar waves simultaneously pass through the same region of space. The resultant displacement at any point is the sum of the individual displacements; the waves emerge from the overlapping region unafafected.

Beats: Regular increase and decrease in the intensity of sound waves caused by the interference of two waves of slightly different frequencies.

Doppler Effect: Apparent change in the frequency of a wave caused by the relative motion between the source and the observer.

Mach Number: Ratio of the speed of a moving object to the speed of sound in the medium at the location where the object is traveling.