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 moving Sound Slinky waves-tr-long
as the wave is Figure 15-5 Figure 15-2 Figure 15-6
Figure 15-10 Wave Forms So far we have examined continuous waves waves that go on forever in each direction!
We can also have pulses caused by a brief disturbance of the medium: v
v v And pulse trains which are somewhere in between.
Figure 15-3 Mathematical Description y Suppose we have some function y = f(x): x
0 y f(x - a) is just the same shape moved a distance a to the right:
Let a = vt Then f(x - vt) will describe the same 0
y v shape moving to the right with speed v. traveling waves properties
x=a x 0 x = vt
x Periodic Wave Properties Wavelength: The distance between identical points on the wave. Amplitude: The maximum displacement A of a point on the wave.
Wavelength Amplitude A A Wave Properties...
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. v T Math... y Consider a wave that is harmonic
in x and has a wavelength of . A x
If the amplitude is maximum at x = 0 this has the functional form: 2 y x A cos x
Now, if this is moving to the right with speed v it will be described by: 2 x vt y x , t A cos
y v x Math...
So we see that a simple harmonic 2
x vt y x , t A cos
wave moving with speed v in the x direction is described by the equation: By using v from before, and by defining
T 2 we can write this as: y x , t A cos kx t (what about moving in the -x direction?)
k 2 Figure 15-7 Math Summary
y The formula y x , t A cos kx t describes a harmonic wave of
amplitude A moving in the +x direction. A x Each point on the wave oscillates in the y direction with
simple harmonic motion of angular frequency . 2 k The wavelength of the wave is
The speed of the wave is v The quantity k is often called the wave number
k v=/T 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! =vT or = 2 v / (sinceT = 2 /
or v / f (since T = 1 / f ) Waves on a string What determines the speed of a wave? Consider a pulse propagating along a string:
v Waves on a string... Suppose: The tension in the string is F
The mass per unit length of the string is (kg/m) The shape of the string at the pulses maximum is
circular and has radius R F R Waves on a string...
Consider moving along with the pulse Apply F = ma to the small bit of string at the top of the pulse v
y x Waves on a string... So we find:
v F v tension F mass per unit length
Making the tension bigger increases the speed. Making the string heavier decreases the speed.
As we asserted earlier, this depends only on the nature of the medium, not on amplitude, frequency, etc. of the wave. 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 Power P = F.v F x
v Power along the string Since v is along the y axis only, to evaluate Power = F.v we only need to find Fy = -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 y x Fy
y ( x , t ) A cos( kx t ) v y x ,t tan dy A sin kx t
dt dy kA sin kx t dx F v dy
dx Recall sin tan cos for small
Power... So: v y x,t Asin kx t kAsin kx t
P(x,t) Fv y kFA 2 sin 2 (kx t ) But last time we showed that v 2 and F v k
P x , t v2 A 2 sin 2 kx t cos kx t sin 2 kx t Average Power We just found that the power flowing past location x on the string at time t is given by:
P x, t v 2 A2 sin2 kx t 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 sin2 (kx - t) is 1/2 and find that: 1
P v 2 A2 2 It is generally true that wave power is proportional to the speed of the wave v and its amplitude squared A2.
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. 1
P 2 A2v We found that 2 dE 1 dx 2 A2
dt 2 dt So 1 dE 2 A2 dx
2 dE 1 2 A2is the average energy per unit length. dx 2
Recap & Useful Formulas: y A x
General harmonic waves y x , t A cos kx t 2 k
2 2 f T v f k
Waves on a string F v
tension mass / length 1 P v 2 A2 2 dE 1 2 A2
dx 2 Superposition Q: What happens when two waves collide? A: They ADD together! We say the waves are superposed.
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 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: A(1t)
B(2t) What does C(t) = A(t) + B(t) look like?? Figure 15-25 (a) Figure 15-25 (b)
Figure 15-25 (c) 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(1t) B(2t) C(t) = A(t) + B(t)
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