What is a Wave Wave is a disturbance

What is a Wave? Wave is a disturbance or variation that transfers energy progressively from point to point in a medium

Transverse Waves In a transverse wave the particle displacement is perpendicular to the direction of wave propagation. The particles do not move along with the wave; they simply oscillate up and down about their individual equilibrium positions as the wave passes by.

Transverse Wave in string

Longitudinal Waves In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions.

Water Waves Water waves are an example of waves that involve a combination of both longitudinal and transverse motions. As a wave travels through the waver, the particles travel in clockwise circles. The radius of the circles decreases as the depth into the water increases.

Transverse and Longitudinal Wave pulse The speed of mechanical wave depends upon the inertia and elasticity of the medium in which wave is travelling. Transverse wave in string travels with a speed, Longitudinal wave in a medium travels with a speed,

Equation of Travelling wave

Equation of Travelling wave

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Solution

Solution

Solution

Sinusoidal Wave Motion in Time and Space • The equation of wave is a function of both space and time expressed by, Y = f(x-vt) Y = f(t-x/v) Y = A sin k(x - vt) Y = A sin w(t - x/v) where A is the amplitude of the wave, ω is the angular frequency of the wave and k is the wave number. The negative sign is used for a wave traveling in the positive x direction and the positive sign is used for a wave traveling in the negative x direction.

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Solution

Solution

Plane Progressive wave x=10. 25 cm Vibration of particle with time at x = 10. 25 cm Snapshot at t = 27 s

The Linear Wave Equation

The Linear Wave Equation

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Solution

The speed of transverse wave v

The speed of transverse wave

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Solution

Illustration

Solution

POWER TRANSMITTED ALONG THE STRING BY A SINE WAVE

POWER TRANSMITTED ALONG THE STRING BY A SINE WAVE

POWER TRANSMITTED ALONG THE STRING BY A SINE WAVE

INTENSITY TRANSMITTED ALONG THE STRING BY A SINE WAVE

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Solution

Superposition of Waves The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time. The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave dispacements. This is true of waves which are finite in length (wave pulses) or which are continuous sine waves.

Superposition & Colliding Waves Principle of Superposition: When two or more waves are simultaneously at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave. Example: Two waves travel on a string in opposite directions. While they overlap, their amplitudes add. However, after the overlap is finished, they proceed unchanged from their initial shapes. Note: If the amplitude grows too large, superposition may fail. In that case, the medium is said to go “non-linear”.

More Waves in Collision Complete cancellation!

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Solution

Reflection of Wave 1. Reflection from a hard boundary Þat a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180 o phase change) The animation at left shows a wave pulse on a string moving from left to right towards the end which is rigidly clamped. As the wave pulse approaches the fixed end, the internal restoring forces which allow the wave to propagate exert an upward force on the end of the string. But, since the end is clamped, it cannot move. According to Newton's third law, the wall must be exerting an equal downward force on the end of the string. This new force creates a wave pulse that propagates from right to left, with the same speed and amplitude as the incident wave, but with opposite polarity (upside down). Animation Equation

Reflection of Pulse from the Fixed Boundary

Reflection from a SOFT boundary Þ at a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave • The animation at left shows a wave pulse on a string moving from left to right towards the end which is free to move vertically. The net vertical force at the free end must be zero. This boundary condition is mathematically equivalent to requiring that the slope of the string displacement be zero at the free end (look closely at the movie to verify that this is true). The reflected wave pulse propagates from right to left, with the same speed and amplitude as the incident wave, and with the same polarity (right-side up). Animation Equation

Reflection of Pulse from the Free Boundary

Reflection and transmission of waves

Refraction of Sound Waves Notice that as the wavefronts cross the boundary the wavelength changes, but the frequency remains constant. The speed of a wave depends on the elastic and inertia properties of the medium through which it travels. When a wave encounters different medium where the wave speed is different, the wave will change directions. Most often refraction is encountered in a study of optics, with a ray of light incident upon a boundary between two media (air and glass, or air and water, or glass and water). Snell's law relates the directions of the wave before and after it crosses the boundary between the two media.

Reflection from an impedance discontinuity Ø When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary. The exact behavior of reflection and transmission depends on the material properties on both sides of the boundary. One important property is the characteristic impedance of the material. The characteristic impedance of a material is the product of mass density and wave speed. Ø In the animations on next slide, two strings of different densities are connected so that they have the same tension. The density of the thick string is 4 times that of the thin string. Ø How do the wave speeds compare for the two strings?

From high speed to low speed (low density to high density) In this animation the incident wave is travelling from a low density (high wave speed) region towards a high density (low wave speed) region ⇒ How do the amplitudes of the reflected and transmitted waves compare to the amplitude of the incident wave? ⇒ How do the polarities of the reflected and transmitted waves compare to the polarity of the incident wave? ⇒ How do the widths of the reflected and transmitted waves compare to the width of the incident wave?

From low speed to high speed (high density to low density) In this animation the incident wave is travelling from a high density (low wave speed) region towards a low density (high wave speed) region. ⇒ How do the amplitudes of the reflected and transmitted waves compare to the amplitude of the incident wave? ⇒ How do the polarities of the reflected and transmitted waves compare to the polarity of the incident wave? ⇒ How do the widths of the reflected and transmitted waves compare to the width of the incident wave?

REFLECTION AND REFRACTION OF WAVE Rarer medium Denser medium Reflected wave suffers a phase change of 1800 from a boundary of denser medium. Denser medium Rarer medium Reflected wave in phase with incident wave reflected from a boundary of rarer medium Ø Notice change in wavelength and amplitude of transmitted and reflected wave in comparison to incident wave.

Illustration

Solution

Solution

Superposition of Two Sinusoidal Waves Travelling in same Direction with varying phase The animation at left shows two sinusoidal waves travelling in the same direction. The phase difference between the two waves varies increases with time so that the effects of both constructive and destructive interference may be seen. First of all, notice that the sum wave (in blue) is a travelling wave which moves from left to right. When the two gray waves are in phase the result is large amplitude. When the two gray waves become out of phase the sum wave is zero.

Constructive & Destructive Interference d= 0 Constructive d= p Destructive

Interference & Path Length

Interference in 2 D

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Solution

Two sine waves travelling in opposite directions create a standing wave A travelling wave moves from one place to another, whereas a standing wave appears to stand still, vibrating in place. Two waves (with the same amplitude, frequency, and wavelength) are travelling in opposite directions on a string.

Standing Wave Algebra

Standing Wave Algebra This wave is no longer a travelling wave because the position and time dependence have been separated. The displacement of the string as a function of position has an amplitude of 2 asin(kx). This amplitude does not travel along the string, but stands still and oscillates up and down according to cos(ωt). Characteristic of standing waves are locations with maximum displacement (antinodes) and locations with zero displacement (nodes).

Standing Waves An example of a standing wave is a vibrating string. It is not obvious, but the standing wave shown (blue) can be considered as the superposition of a wave traveling to the left (red) and one traveling to the right (green). As the phases of the waves change, they alternately reinforce and cancel. The maxima of the standing wave occur at points called “antinodes”, and the zeroes occur at points called “nodes”.

Nodes and Antinodes If we plot all the snapshots together, we get the figure shown at the right. There are points called nodes than have zero displacement. These are spaced a distance l/2 apart, and they include the point at which the string is anchored to the wall. These are points of maximum destructive interference. Between the nodes are points of maximum displacement. These are called antinodes. These are also spaced l/2 apart and are points of maximum constructive interference.

Intensity vs. Position The intensity (energy content) of a wave is proportional to the amplitude squared: I = CA 2. The figure shows that the points of maximum intensity correspond to the antinodes, and the points of zero intensity to the nodes. Therefore, maximum intensity corresponds to constructive interference, and zero intensity corresponds to complete destructive interference.

Formation of Standing Wave Ø Notice the superposition of incident and reflected wave (reflected from a denser boundary) and formation of standing wave.

Animation on standing wave If two sinusoidal waves having the same frequency (wavelength) and the same amplitude are travelling in opposite directions in the same medium then, using superposition, the net displacement of the medium is the sum of the two waves. When the two waves are 180° out-ofphase with each other they cancel, and when they are inphase with each other they add together. As the two waves pass through each other, the net result alternates between zero and some maximum amplitude. However, this pattern simply oscillates; it does not travel to the right or the left. Two dots on the string, one at an antinode and one at a node.

Standing Wave in String Fixed at both the ends Standing wave in first harmonic, Wavelength = 2 m Standing wave in second harmonic, Wavelength = 1 m Standing wave in third harmonic, Wavelength = 0. 5 m

Closed on both ends (Piano)

Piano – Guitar Notes are just standing waves of a specific frequency.

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Solution

Solution

Illustration The following equations represent transverse waves; z 1 = A cos (kx – wt) z 2 = A cos (kx + wt) z 3 = A cos (ky – wt) Identify the combination (s) of the waves which will produce (i) standing wave (s), (ii) a wave travelling in the direction making an angle of 45 degrees with the positive x and positive y-axes. In each case, find the position at which the resultant intensity is always zero.

Solution

Solution

Reflection from Closed end of Organ Pipe One cycle of a sinusoidal pressure wave (e. g. a sound wave): (a)approaching the closed end of an air column; (b) returning after reflection from the closed end

Reflection from Open end of Organ Pipe One cycle of a sinusoidal pressure wave (e. g. a sound wave): (a)approaching the open end of an air column; (b) returning after reflection from the open end

Standing Waves in Organ Pipe Open at Both Ends

Waves in a Closed-Closed Pipe A long narrow column of air such as the air in a tube or pipe can support a longitudinal standing sound wave. Such a tube may be open or closed at each end. A closed end causes a node there. An open end causes an antinode. In the example shown here, both ends are closed and the standing wave mode is m=2. There are nodes at each end and one in the center, and there are two antinodes at the quarter wave locations.

Waves in an Open-Open Pipe n

Waves in an Open-Closed Pipe

Pipes and Modes Open-Open or Closed-Closed Open-Closed

Tube Open on both ends (Flute)

Tube Open – Closed (Pipe organ) Only odd harmonics n = 1, 3, 5, 7, … (We need a node at the closed end an antinode at the open end)

Two guassian waves travelling in opposite directions The movie at left shows two gaussian wave pulses are travelling on a string, one is moving to the right, the other is moving to the left. They pass through each other without being disturbed, and the net displacement is the sum of the two individual displacements. It should also be mentioned that this string is nondispersive (all frequencies travel at the same speed) since the Gaussian wave pulses do not change their shape as they propagate. If the medium was dispersive, then the waves would change their shape.

Two sine waves with different frequencies: Beats Two waves of equal amplitude are travelling in the same direction. The two waves have different frequencies and wavelengths, but they both travel with the same wave speed. Using the principle of superposition, the resulting particle displacement can be found as shown below. •

Beats and Modulation This resulting particle motion is the product of two travelling waves. One part is a sine wave which oscillates with the average frequency, f = ½(f 1 + f 2), This is the frequency which is perceived by a listener. The other part is a cosine wave which oscillates with the difference frequency f = ½(f 1 - f 2). This term controls the amplitude "envelope" of the wave and causes the perception of "beats". The beat frequency is actually twice the difference frequency, fbeat = (f 1 - f 2).

Animation on Beats In the movie at left two waves with slightly different frequencies are travelling to the right. The resulting wave travels in the same direction and with the same speed as the two component waves. The "beat" wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency.

Graphical Beats

Standing Wave In Hanging Chain

Propagation of spherical wave pulse in a homogeneous medium In the animation a spherical wave pulse propagates in a medium where the wave speed is constant in all directions. The wave expands outwards as an ever expanding circle, with the wave traveling at the same speed in all directions. Since the wave speed is the same everywhere, there is no refraction, and the wave does not change direction as it propagates.

Propagation of spherical wave pulse in a non - homogeneous medium The animation shows the propagation of a spherical wave pulse in a medium where the wave speed in the x-direction is constant, but where the speed in the vertical ydirection decreases with height (c = 1 - 0. 05 y). As a result the wave changes direction and bends upwards.

Propagation of spherical wave pulse in a non - homogeneous medium The animation above shows the propagation of a spherical wave pulse in a medium where the wave speed in the x-direction is constant, but where the speed in the vertical y-direction increases with height (c = 1 + 0. 05 y). As a result the wave changes direction and bends downwards.

Source of sound The source of sound is a vibrating object. Standing waves are produced in musical instruments by striking, blowing, plucking, bowing • Pitch - Frequency • Amplitude - Loudness (Energy) • Sound Quality - Wave form • Vibrating strings (Violin, Guitar) Vibrating columns of air (Flute, trumpet, pipe organ)

Sound intensity in d. B

d. B Calculations

Sound Color- Timbre