Waves and Sound

Wave MotionA wave is a moving self-sustained

disturbance of a medium – either a field or a substance.

Mechanical waves are waves in a material medium.

Mechanical waves requireSome source of disturbanceA medium that can be disturbedSome physical connection between or

mechanism though which adjacent portions of the medium influence each other

All waves carry energy and momentum

Wave CharacteristicsThe state of being displaced moves

through the medium as a wave.A progressive or travelling wave is a self-

sustaining disturbance of a medium that propagates from one region to another, carrying energy and momentum.

Examples: waves on a string, surface waves on liquids, sound waves in air, and compression waves in solids or liquids.

In all cases the disturbance advances and not the medium.

Traveling Waves

Flip one end of a long rope that is under tension and fixed at one end

The pulse travels to the right with a definite speed

A disturbance of this type is called a traveling wave

Description of a WaveA steady stream of

pulses on a very long string produces a continuous wave

The blade oscillates in simple harmonic motion

Each small segment of the string, such as P, oscillates with simple harmonic motion

Amplitude and Wavelength

Amplitude (A) is the maximum displacement of string above the equilibrium position

Wavelength (λ), is the distance between two successive points that behave identically

Longitudinal Waves

In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave

A longitudinal wave is also called a compression wave

Longitudinal Wave Represented as a Sine Curve

A longitudinal wave can also be represented as a sine curve

Compressions correspond to crests and stretches correspond to troughs

Also called density waves or pressure waves

Transverse Waves

In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion

WaveformsWavepulse in taut rope.

Shape of pulse is determined by motion of driver.

If driver (hand) oscillates up and down in a regular way, it generates a wave train – a constant frequency carrier whose amplitude is modulated (varies with time.)

Waveform – The shape of a Wave

The high points are crests of the wave

The low points are troughs of the wave

As a 2-D or 3-D wave propagates, it creates a wavefront

Velocity of WavesPeriod (T) of a periodic wave - time it takes for

a single profile to pass a point in space - the number of seconds per cycles.

The inverse of the period (1 /T) is the frequency f, the number of profiles passing per second, the number of cycles per second.

The distance in space over which the wave executes one cycle of its basic repeated form is the wavelength, – the length of the profile.

Velocity of WavesThe speed of the wave — the rate (in m/s) at

which the wave advancesIs derived from the basic speed equation of

distance/timeSince a length of wave passes by in a time

T, its speed must equal /T = f The speed of any progressive periodic wave:

v = f

Example 1 A youngster in a boat watches waves on a

lake that seem to be an endless succession of identical crests passing, with a half-second between them. If one wave takes 1.5 s to sweep straight down the length of her 4.5 m-long boat, what are the frequency, period, and wavelength of the waves?

Given: The waves are periodic; 0.5 s between crests; L = 4.5 m; t = 1.5 s

Find: T, f, v, and

Transverse Waves: StringsThe speed of a mechanical wave is determined by

the inertial and elastic properties of the medium and not in any way by the motion of the source

Pulse traveling with a speed v along a lightweight, flexible string under constant tension FT

v (11.3)

When m/L is large, there is a lot of inertia and the speed is low. When FT is large, the string tends to spring back rapidly, and the speed is high

Lm

FT/

Example 2A 2.0 m-long horizontal string having a mass

of 40 g is slung over a light frictionless pulley, and its end is attached to a hanging 2.0 kg mass. Compute the speed of the wavepulse on the string. Ignore the weight of the overhanging length of rope.

Given: A string of length l = 2.0 m, m = 40 g supporting a 2.0 kg load

Find: v

Reflection, Refraction, Diffraction and Absorption

End of rope is held stationary; energy pumped in at the other end, the reflected wave ideally carries away all the original energyIt is inverted – 180° out-of-phase with the

incident waveEnd of the rope is free; it will rise up as the pulse

arrives until all the energy is stored elastically. The rope then snaps back down, producing a

reflected wavepulse that is right side up.

Reflection of Waves – Fixed Boundary

Whenever a traveling wave reaches a boundary, some or all of the wave is reflected

When it is reflected from a fixed end, the wave is inverted

The shape remains the same

Reflected Wave – Open Boundary

When a traveling wave reaches an open boundary, all or part of it is reflected

When reflected from an open boundary, the pulse is not inverted

When a wave passes from one medium to another having different physical characteristics, there will be a redistribution of energy. Medium is also displaced, and a portion of the incident

energy appears as a refracted wave. If the incident wave is periodic, the transmitted wave has

the same frequency but a different speed and therefore a different wavelength: the larger the density of the refracting medium, the smaller the length of the wave.

Reflection, Refraction, Diffraction and Absorption

When a wave meets a hole or another obstacle, it can be bent around it or through it—Diffraction

A wave can lose part or all of its energy when it meets a boundary – Absorption.

Reflection, Refraction, Diffraction and Absorption

A wave passing through a “lens” will be both reflected AND refracted. Examples include light (of course) and also sound (through the balloon of different gas)

Absorption can either SUBTRACT (beach sand) or ADD (wind) energy to a wave, depending on which way the energy is being transferred.

Reflection, Refraction, Diffraction and Absorption

Superposition of WavesSuperposition Principle: In the region

where two or more waves overlap, the resultant is the algebraic sum of the various contributions at each point. Superimposing two harmonic waves of the same

frequency and amplitude: at every value of x, add the heights of the two sine curves – above the axis as positive and below it as negative.

The sum of any number of harmonic waves of the same frequency traveling in the same direction is also a harmonic wave of that frequency.

Interference of WavesTwo traveling waves can meet and pass

through each other without being destroyed or even altered

Waves obey the Superposition PrincipleIf two or more traveling waves are moving

through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point

Actually only true for waves with small amplitudes

Constructive Interference

Two waves, a and b, have the same frequency and amplitudeAre in phase

The combined wave, c, has the same frequency and a greater amplitude

Destructive Interference

Two waves, a and b, have the same amplitude and frequency

They are 180° out of phase

When they combine, the waveforms cancel

SuperpositionWhen two or more waves interact, their amplitudes are added (superimposed) one upon the other, creating interference.

Constructive interference occurs when the superposition increases amplitude.Destructive interference occurs when the superposition decreases the amplitude.

Natural Frequency/HarmonicsIf a periodic force occurs at the appropriate frequency, a standing wave will be produced in the medium.

The lowest natural frequency in a medium is its

fundamental harmonic.

Double this frequency to produce the 2nd harmonic.

Triple this frequency to produce the 3rd harmonic

REQUIRES FIXED BOUNDARIES

Frequency and Period

0 - the natural angular frequency, the specific frequency at which a physical system oscillates all by itself once set in motion

natural angular frequency

and since 0 = 2f0

natural linear frequency

Since T= 1/f0

Period

m

k0

m

kf

2

10

k

mT 2

Waves and EnergyAs waves propagate, their energy alternates between two froms:

Transverse Waves – Potential <> KineticLongitudinal Waves – Pressure <> KineticLight Waves – Electric <> Magnetic

Generally –

HIGHER FREQUENCY = HIGHER ENERGY

HIGHER AMPLITUDE = HIGHER ENERGY

Waves and Energy

Nodes and Modes Nodes occur/are located at points of

equilibrium within a wave.

Anitnodes occur/are located at points of greatest displacement (amplitude) within a wave.

One-dimensional modes:Transverse – guitar or piano stringsRotational – jump rope, lasso

Two- and Three-dimensional modes:Radial – concentric circular nodes and anti-

nodesAngular – linear nodes and anti-nodes

radiating outward from center.

Nodes and Modes

Nodes and Modes