Physics 260 Conceptual discussion of wave motion Wave properties Mathematical description Waves on a string

What is a wave ? : A wave is a traveling disturbance that transports energy but not matter. Examples: –Sound waves (air moves back & forth) –Stadium waves (people move up & down) –Water waves (water moves up & down) –Light waves (what moves??)

Characteristics of Wave Motion –Where do Waves come from and what are Periodic Waves? waves in mechanical media. Wave types –Transverse and Longitudinal Energy transported by Waves –How do waves Transmit Energy? Mathematical representation of a traveling wave The Principle of Superposition Interference Standing Waves: Resonances

Types of Waves Transverse: The medium oscillates perpendicular to the direction the wave is moving. –Water –Slinky Longitudinal: The medium oscillates in the same direction as the wave is moving –Sound –Slinky waves-tr-longwaves-tr-long

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Wave Forms continuous wavesSo far we have examined “continuous waves” that go on forever in each direction! v v l We can also have “pulses” caused by a brief disturbance of the medium: v l And “pulse trains” which are somewhere in between.

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Mathematical Description Suppose we have some function y = f(x): x y l f(x - a) is just the same shape moved a distance a to the right: x y x = a 0 0 l Let a = vt Then f(x - vt) will describe the same shape moving to the right with speed v. traveling waves properties x y x = vt 0 v

Periodic Wave Properties Wavelength Wavelength: The distance between identical points on the wave. Amplitude A l Amplitude: The maximum displacement A of a point on the wave. A

Wave Properties... l Period: The time T for a point on the wave to undergo one complete oscillation. Speed: The wave moves one wavelength in one period T so its speed is v =  / T.

Math... Consider a wave that is harmonic in x and has a wavelength of. If the amplitude is maximum at x = 0 this has the functional form: y x A l Now, if this is moving to the right with speed v it will be described by: y x v

Math... l By usingfrom before, and by defining l So we see that a simple harmonic wave moving with speed v in the x direction is described by the equation: we can write this as: (what about moving in the -x direction?)

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Math Summary l The formula describes a harmonic wave of amplitude A moving in the +x direction. y x A Each point on the wave oscillates in the y direction with simple harmonic motion of angular frequency . l The wavelength of the wave is l The speed of the wave is l The quantity k is often called the “wave number”

Wave Properties... We will show that the speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength, or period. and T are related! = v T or = 2  v /  (since  T = 2  /   or  v / f (since T = 1 / f )  v = / T

Waves on a string What determines the speed of a wave? Consider a pulse propagating along a string: v

Waves on a string... l The tension in the string is F The mass per unit length of the string is  (kg/m) l The shape of the string at the pulse’s maximum is circular and has radius R R  F Suppose:

Waves on a string... v x y l Consider moving along with the pulse l Apply F = ma to the small bit of string at the “top” of the pulse

Waves on a string... l So we find: l Making the tension bigger increases the speed. l Making the string heavier decreases the speed. l As we asserted earlier, this depends only on the nature of the medium, not on amplitude, frequency, etc. of the wave. v tension F mass per unit length 

Wave Power A wave propagates because each part of the medium communicates its motion to adjacent parts. –Energy is transferred since work is done! How much energy is moving down the string per unit time. (i.e. how much power?) P

Wave Power... Think about grabbing the left side of the string and pulling it up and down in the y direction. You are clearly doing work since F. dr > 0 as your hand moves up and down. This energy must be moving away from your hand (to the right) since the kinetic energy (motion) of the string stays the same. P

How is the energy moving? Consider any position x on the string. The string to the left of x does work on the string to the right of x, just as your hand did: x  x F Power P = F. v v

Power along the string Since v is along the y axis only, to evaluate Power = F. v we only need to find F y = -F sin   -F  if  is small. We can easily figure out both the velocity v and the angle  at any point on the string: If  x F v y Recall sin   cos   for small   tan  FyFy  dy dx

Power... So: l But last time we showed that and

Average Power We just found that the power flowing past location x on the string at time t is given by: l We are often just interested in the average power moving down the string. To find this we recall that the average value of the function sin 2 (kx -  t) is 1 / 2 and find that: l It is generally true that wave power is proportional to the speed of the wave v and its amplitude squared A 2.

Energy of the Wave We have shown that energy “flows” along the string. –The source of this energy (in our picture) is the hand that is shaking the string up and down at one end. –Each segment of string transfers energy to (does work on) the next segment by pulling on it, just like the hand. l We found that Sois the average energy per unit length.

Recap & Useful Formulas: y x A l Waves on a string l General harmonic waves tension mass / length

Superposition Q:Q: What happens when two waves “collide?” A:A: They ADD together! –We say the waves are “superposed.”

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Aside: Why superposition works As we will see shortly, the equation governing waves (a.k.a. “the wave equation”) is linear. –It has no terms where variables are squared. x = B sin(  t) + C cos(  t) For linear equations, if we have two (or more) separate solutions, f 1 and f 2, then Bf 1 + Cf 2 is also a solution! You have already seen this in the case of simple harmonic motion: linear in x!

Superposition & Interference We have seen that when colliding waves combine (add) the result can either be bigger or smaller than the original waves. We say the waves add “constructively” or “destructively” depending on the relative sign of each wave. will add constructively will add destructively l In general, we will have both happening

Superposition & Interference Consider two harmonic waves A and B meeting at x=0. –Same amplitudes, but  2 = 1.15 x  1. The displacement versus time for each is shown below: What does C(t) = A(t) + B(t) look like?? A(  1 t) B(  2 t)

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Interference Interference: Combination of two or more waves to form composite wave –use superposition principle. Waves can add constructively or destructively Conditions for interference: 1.Coherence: the sources must maintain a constant phase with respect to each other 2.Monochromaticity: the sources consist of waves of a single wavelength

Superposition & Interference Consider two harmonic waves A and B meeting at x = 0. –Same amplitudes, but  2 = 1.15 x  1. The displacement versus time for each is shown below: A(  1 t) C(t) = A(t) + B(t) CONSTRUCTIVE INTERFERENCE DESTRUCTIVE INTERFERENCE B(  2 t)

Beats l Can we predict this pattern mathematically? çOf course! l Just add two cosines and remember the identity: whereand cos(  L t)

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Chapter 16 Sound Physics for Scientists & Engineers, 3 rd Edition Douglas C. Giancoli © Prentice Hall

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