Wave Interference with Applications



Wave Interference with Applications

Math Aside:

The following trigonometry identity will be useful for this topic: sinα + sinβ = 2cos[(α-β)/2] sin[(α+β)/2]

The superposition principle shows how to find the wave function when more than one wave occupies a string. We call this wave interference, wave addition or wave superposition; these all mean simply, add the waves,

y = y1 + y2 .

Applications

One result of this principle is the phenomenon of constructive and destructive interference. We will also examine two more applications: standing waves and beats.

I. Adding two identical sinusoidal traveling waves differing only by a phase constant, δ

A. One dimension

Using the trig. identity and superposition principle above,

y = A sin(kx - ωt) + A sin(kx - ωt + δ)

y = 2A cos[δ/2] sin[kx - ωt +δ/2]

Special cases:

δ = 0 ( constructive interference

Corresponding points line up, i.e. crests line up with crests, and troughs line up with troughs.

δ ’ π ( destructive interference

From the trig. identity below we see this is like a λ/2 translation of one of the waves.

sin(C + δ) = sinC cosδ + cosC sinδ

= -sinC (δ=π was assumed)

Comment: We see that the result of this addition is a sinusoidal traveling wave with the average phase shift and a modified amplitude. Each point of the string is in SHM with the same frequency as before.

B. Two dimensions

When two point sources of waves interfere at a point P the interference conditions may be written in terms of the path difference traveled by the two waves, Δr, using Δr/δ’λ/2π

Examples of this are the famous Young’s double slit interference experiment and also examples are common in acoustics with speakers. Place below the detailed derivation given in class.

δ=n2π or Δr=n λ n=0,1,2. [construc. interf.]

δ=(n+1/2)2π or Δr=(n +1/2)λ n=0,1,2….[destruc. interf.]

II. Adding two identical sinusoidal traveling waves differing only by their direction of motion: standing waves

Using the trig. identity and superposition principle above,

y = Asin(kx - ωt) + Asin(kx + ωt)

y = 2Asinkx cosωt

This is an altogether new thing, it is not a traveling wave but is called a standing wave. Each point of the string is in SHM with the same angular vibration frequency, ω, and amplitude that depends on the place x. These are called normal modes of vibration.

Points for which sinkx is zero are called nodes, they never move. Antinodes are places of maximum sinkx and therefore vibrate with amplitude 2A.

If both ends of the string is a node then kL must equal an integer times π and therefore, L = nλ/2 n=1,2,3… and using v=fλ gives only the following frequencies will generate a standing wave pattern,

fn = nf1 , where the fundamental is given by f1=v/2L.

III. Adding two identical sinusoidal traveling waves differing only by frequency: beats

Using the superposition principle,

y = Asin[kx-ωt] + Asin[k’x-ω’t] = A[sinωt +sinω’t]

where I’ve assumed x=0 in the last step. The trig. identity

gives,

y = 2Acos[(ω’-ω)t/2] sin[(ω’+ ω)t/2].

A graph of this gives an envelope inside of which a rapidly varying function is enclosed. The beat period is half that of the envelope.

This leads to a beat frequency of

fbeat = f’ – f .

EXAMPLES [in class]

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