How Long for Wave Amplitude to Decrease by a Factor of e?

[knowlewj01]

Homework Statement

A wave is driven at z=0 with constant real frequency ω_r propagates in the z direction, for z>0 the amplitude varies as:

A = A_0 e^{i\omega_r - ikz}

where k is complex
k=k_r - i k_i

if a wave with spatially constant amplitude and purely real wavenumber k_r were excited in the same medium, how long would pass before the wave's amplitude decreased by a factor of e?

Homework Equations

The Attempt at a Solution

(I'm not too sure what is meant by 'spatially constant')

does anyone know where to start on this one? I've tried a few things like removing the i from ikz, removing the z dependence altogether, I'm having trouble picturing what's going on.

 

[ardie]

if a wave has spatially constant amplitude, means its amplitude is a constant and independent of space directional variables.

now any wave that decays, does so due to some restrain which is put on it from its medium by some means; so we must seek a decaying solution that satisfies the above wave propagation equation, where the amplitude decays as a result of the given equation of waves for the medium.

 

[knowlewj01]

The simplest waveform i can think of that has constant amplitude is

x = x_0 cos(\omega t)

but wouldn't a decay indicate behaviour more like

x = x_0 e^{-\frac{t}{\tau}}cos(\omega t)

but then how does the wavenumber come into it?

 

[ardie]

that expression is very correct, but the incorrect approach to solve the problem, try writing the wave function in complex exponential notation and insert the wave number with the imaginary part in there.

 

[olivermsun]

Your simplest waveform has not only spatially constant amplitude but in fact has no spatial (x) dependence at all (it doesn't propagate!).

Think about the "excited wave" as an initial condition for a general wave propagating in the medium and then think about the disturbance evolves in time.

As ardie pointed out, complex exponentials are your friend.

 

[knowlewj01]

x=x_0 e^{i\omega t - ikz} \hat{x}

is the general form of a plane wave, this would propagate in the z direction with constant amplitude. correct?

so initially the wavevector is complex, k = kr - i ki

when the wavevector is only real, such that k = kr

x_i = x_0e^{i\omega t - (ik_r + k_i)z}
x_f = x_0e^{i\omega t - ik_rz}

im going to eliminate t if i carry on this way. is this any closer?

 

[olivermsun]

You know that the wave amplitude must have both space and time dependence, since they tell you the spatial dependence and they ask you about the time dependence. Thus you might conclude that your wave must have a form like x = A(z,t) exp(iωt - ikz).

Your first expression tells you the decay of the wave amplitude as it propagates in the +z direction, but notice that the expression has no time dependence. You also have an initial condition which tells you what the original amplitude and wavenumber are (what is t at the initial condition?). So how do you get the time dependence of a wave if you know the spatial dependence?

 

[knowlewj01]

ive just noticed i made a typo in the question

A = A_0e^{i\omega_r - ikz}

should read

A = A_0e^{i\omega_rt - ikz}

so there should be a time dependance.

sorry