The Principle of Superposition

The Principle of Superposition For small amplitudes, waves will pass through each other and emerge unchanged. Superposition Principle: When two or mo...
Author: Pauline Lane
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The Principle of Superposition For small amplitudes, waves will pass through each other and emerge unchanged. Superposition Principle: When two or more waves overlap, the net disturbance at any point is the sum of the individual disturbances due to each wave.

Overlapping waves algebraically add to produce a resultant wave (or net wave). Overlapping waves do not in any way alter the travel of each other.

Two traveling wave pulses: left pulse travels right; right pulse travels left.

Superposition y1 = A sin (k1 x - w1 t) and y2 = A sin (k2 x - w2 t + ϕ). Since sin (a) + sin (b) = 2 cos ((a - b) / 2) sin ((a + b) / 2), we can form the sum y=y1+y2 = 2 A cos [(Δk x - Δw t + ϕ) / 2] sin [(Σk x - Σw t + ϕ) / 2], where Δk is k 2 - k 1, Δw is w 2 - w 1, Σk is k 1 + k 2 and Σw is w 1 + w 2

Interference and Diffraction Two waves are considered coherent if they have the same frequency and maintain a fixed phase relationship. Two waves are considered incoherent if the phase relationship between them varies randomly. When waves are in phase, their superposition gives constructive interference. When waves are one-half a cycle out of phase, their superposition gives destructive interference.

For "coherent sources", in which the two waves have the same wave number and frequency (k 1 = k 2, w 1 = w 2): y(x,t) = 2 A cos (ϕ / 2) sin (k x - w t + ϕ / 2). Amplitude

…..back to coherent waves

When two coherent waves travel different distances to reach the same point, the phase difference is determined by:

d1 − d 2

λ

phase difference = 2π

http://www.falstad.com/ripple/

We can represent wave vectorially with a phasor.

We can use phasors to combine waves even if their amplitudes are different.

For the horizontal components we have

For the vertical components we have

Thus, the resultant wave has an amplitude of

and a phase constant of

Beats When two waves with the same amplitude but different frequency are added together a phenomenon called "beating" y2 = Asin(2πf2t) occurs. y1 = Asin(2πf1t)

The resultant waveform can be thought of as a wave with frequency fave = (f1 + f2)/2 which is constrained by an envelope with a frequency of fb = |f1 - f2|. The envelope frequency is called the beat frequency. http://library.thinkquest.org/19537/java/Beats.html

Standing Waves Pluck a stretched string such that y(x,t) = A sin(ωt + kx) When the wave strikes the wall, there will be a reflected wave that travels back along the string. If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave.

The reflected wave will be 180° out of phase with the wave incident on the wall. Its form is y(x,t) = -A sin (ωt - kx). Apply the superposition principle to the two waves on the string:

y ( x, t ) = y1 ( x, t ) + y2 ( x, t )

= A(sin (ωt + kx ) − sin (ωt − kx )) = (2 A cos ωt )sin kx

The previous expression is the mathematical form of a standing wave. A

A

A

N

N N

N

A node (N) is a point of zero oscillation. An antinode (A) is a point of maximum displacement. All points between nodes oscillate up and down. http://www.falstad.com/loadedstring/

http://www.falstad.com/membrane/

The nodes occur where y(x,t) = 0.

y ( x, t ) = 2 A cos ωt sin kx = 0 The nodes are found from the locations where sin kx=0, which happens when kx = 0, π, 2π,…. That is when kx = nπ where n = 0,1,2,… The antinodes occur when sin kx=± 1; that is where

kx =

π 3π ,

,…

2 2 ( 2n + 1)π kx = and n = 0 ,1, 2 ,… 2

If the string has a length L, and both ends are fixed, then y(x=0,t) = 0 and y(x=L, t) = 0.

y ( x = 0, t ) ∝ sin k (0 ) = 0 y ( x = L, t ) ∝ sin kL = 0 kL = nπ 2π

λ The wavelength of a standing wave:

L = nπ

2L λ= n

where n = 1, 2, 3,…

2L λn = n

These are the permitted wavelengths of standing waves on a string; no others are allowed.

The speed of the wave is:

v = λn f n

The allowed frequencies are then: f n =

v

nv = λn 2 L

n =1, 2, 3,…

The n=1 frequency is called the fundamental frequency. nv ⎛ v ⎞ fn = = = n⎜ ⎟ = nf1 λn 2 L ⎝ 2L ⎠ v

All allowed frequencies (called harmonics) are integer multiples of f1.

Example (text problem 11.55): A Guitar’s E-string has a length 65 cm and is stretched to a tension of 82 N. It vibrates with a fundamental frequency of 329.63 Hz. Determine the mass per unit length of the string. For a wave on a string:

v=

F

μ

Solving for the linear mass density:

F F F μ= 2 = = 2 2 v (λ1 f1 ) f1 (2 L )2

( 82 N ) = (329.63 Hz )2 (2 * 0.65 m )2

= 4.5 × 10 − 4 kg/m