Wave Propagation Homework: Confirm Wave Equation & Obtain Velocity

[EmmaK]

Homework Statement

Consider
y_1=Asin(5x)exp(-2t)
y_2=Aexp(4ix)exp(-2it)
y_3=Asin(2x-5t)exp(-2t)

(i)which one represents a wave that propagates at constant speed with no change in its profile
(ii)Confirm it satifies the wave equation and obtain the wave velocity
(iii)Comment qualitatively on the behaviour descried by the other 2

Homework Equations

The Attempt at a Solution

(i) i think it's the second one. I took the Re(y) and got Acos(3x-2t) so the amplitude is constant, while the other 2 will decrease in amplitude as exp(-2t) will tend to 0?

(ii)just differeniate it and stick into equation

(iii)They will oscillate with the same frequency but the amplitude will decrease?

 

[EmmaK]

for (ii) i have just differentiated and subbed it into the wave equation and got v= 0.5i. can you have and imaginary wave speed?

 

[vela]

for (ii) i have just differentiated and subbed it into the wave equation and got v= 0.5i. can you have and imaginary wave speed?

No, you must have dropped a factor of i somewhere.

Edit: Actually, you probably flipped a sign somewhere since the wave equation tells you what v^2 is.

 

[vela]

Consider
y_1=Asin(5x)exp(-2t)
y_2=Aexp(4ix)exp(-2it)
y_3=Asin(2x-5t)exp(-2t)

(iii)They will oscillate with the same frequency but the amplitude will decrease?

Do you know what the effect is of the -5t in y3, in comparison to y1?

 

[EmmaK]

Do you know what the effect is of the -5t in y3, in comparison to y1?

not really. isn't 5 the frequency? actually no.. because that would mean y1 has a frequency of 0...

 

[vela]

not really. isn't 5 the frequency? actually no.. because that would mean y1 has a frequency of 0...

Actually, you're right. It's the frequency, so the first one doesn't oscillate at all. I'm not sure I'd call it a wave. The exponential factor in y1 and y3, as you mentioned above, just causes the amplitude to decrease over time.

 

[EmmaK]

ahhh ok thanks, so which doesn't change its profile. i thought 2 as it would keep repeating, but the actual height would be constantly changing

 

[vela]

Oh, I hope you don't mean you're changing your answer to (i). You're right that the answer to (i) is y2.

The reason I asked about -5t was because in your answer to (iii) you said y1 and y3 both oscillated with the same frequency. Y3 oscillates, but y1 doesn't.

 

[kiranm]

is the Re(y2) not A cos(4x-2t) instead of A cos(3x-2t) (posted above) cus how would you get to that answer?

 

[vela]

Yes, you're right. I think it was just a typo on Emma's part.

 

[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?

 

[kiranm]

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?

 

[vela]

What's the difference in how e^iωt, e^-iωt, e^-ωt, and e^ωt behave? (Assume ω>0.) In particular, what do these functions do as t goes to ±∞?