Take the real part of complex wave function

[kiranm]

Homework Statement

how do I take the real part of y2= A exp (4ix) exp (-2it)? And how does this determine that this wave propagates with constant speed compared to these wave disturbances:
y1= A sin (5x) exp (-2t)
y3= A sin (2x-5t) exp (-2t)

Homework Equations

exp(ix)= cos x + i sin x

The Attempt at a Solution

For Re(y2) i got A cos (4x) cos (-2t). I don't think that is correct i just don't know how to combine that because i know taking the real part is the cos but my attempt doesn't make sense.

 

[CompuChip]

Let's start with the real part question.

You are right that exp(ix) = cos(x) + i sin(x), so you can plug that into the equation:
exp(4ix) exp(-2it) = (... + ... i)(... + ... i)
When work out the brackets, so you can write it in the form
Re(y2) + Im(y2) i

Or you can first combine the exponentials to exp(4ix) exp(-2it) = exp(...) and then use the identity.

 

[kiranm]

so would it be exp(4ix-2it) = exp i(4x-2t)= cos (4x-2t) + i sin (4x-2t)?

 

[kiranm]

but I am not understanding how this tells u that the wave propagates at constant speed with no change in its profile compared to the other two wave disturbances?

and how can u tell that y1 is a stationary wave whose amplitude is decreasing exponentially with time and that y3 is a traveling wave also decreasing exponentially with time?

 

[ideasrule]

Examine the angle 4x-2t. If that stays the same, y2 obviously stays the same as well. So, what's required for 4x-2t to keep constant as t changes? Can you prove that if x=x0+vt for any x0 and some v, then 4x-2t is a constant? What does "v" have to be?

Once you prove that, the second part:

and how can u tell that y1 is a stationary wave whose amplitude is decreasing exponentially with time and that y3 is a traveling wave also decreasing exponentially with time?

should be easy. Examine the sinuisoidal part of y1; what's v? Do the same for y3, then consider what effect the exponential term has on the wavefunction.