Calculate the velocity when 2 progressive waves are added together

Bolter · Mar 6, 2020

So here is my problem

Text problem: two progressive waves y1 = A sin(2πx/λ − ωt) and y2 = A sin(2πx/λ − ωt − π/2) travel in the same direction; calculate the amplitude Atot of y = y1 + y2, with A = 5.9 cm, λ = 4 m, ω = 10π Hz.

I have had a go at this and get an answer of 8.34cm to be my final amplitude

Handwritten derivation of two sinusoidal waves y1 = A sin(2π/λ x − ωt) and y2 = A sin(2π/λ x − ωt − π/2) being summed using the sine addition identity, giving y1 + y2 = 2A cos(π/4) sin(2π/λ x − ωt − π/4).

Handwritten working on lined paper: A_tot = 2A cos(π/4), then = 2(5.9) cos(π/4), then = 8.34 cm.

The next part that I have been given is to calculate the velocity of when 2 waves are added together. I'm not so sure how to go about this, this is what I have tried anyway

Printed text problem: two progressive waves y1 = A sin(2π/λ x − ωt) and y2 = A sin(2π/λ x − ωt − π/2) travel in the same direction; take A = 5 cm, λ = 7.0 m and ω = 17.4 rad/s, and find the velocity of the interference wave in SI units.

Handwritten derivation: y1 = A sin(2π/λ x − ωt) and y2 = A sin(2π/λ x − ωt − π/2) summed to give 2A cos(π/4) sin(2π/λ x − ωt − π/4); then substituted values to y(x) = 7.07 sin(−17.4t + 2π/7 x − π/4) and v = dy/dt = −123.018 cos(−17.4t + 2π/7 x − π/4).

Can someone please help on how to do work out the velocity of the resultant wave?

Thanks!

Bolter · Mar 6, 2020

Could I just work out the velocity of the wave produced from doing this?

Handwritten derivation: omega = 2 pi f, so f = omega/(2 pi); v = f lambda, so v = omega lambda/(2 pi); then v = (17.4 x 7)/(2 pi) = 19.4 m s^-1.

I'm really confused here

Bolter · Mar 6, 2020

So I have thought about this question again, and I'm thinking that they are asking for the maximum velocity here of the wave that gets produced, since I cannot work out the velocity at any given time of the wave as I am not given a time value or the horizontal displacement x

So would this be an appropriate thing to do?

Handwritten derivation of a standing wave's maximum velocity: y(x) = 2A cos(π/4) sin(2πx/λ − ωt − π/4), rewritten with A_new = 2A cos(π/4), velocity v = dy/dt, and |v_max| = A_new ω = 1.23 m/s.

Bolter · Mar 6, 2020

If a speed of 1.23 m/s is correct? Would it be best to include the negative sign as it asks for the velocity so I shouldn't be taking the absolute value of that then

haruspex · Mar 6, 2020

Your post #2 looks right.
In post #3 you have calculated the max velocity at which a point on the wave rises. That is not the velocity of the wave.

Bolter · Mar 7, 2020

haruspex said:

Your post #2 looks right.
In post #3 you have calculated the max velocity at which a point on the wave rises. That is not the velocity of the wave.

Ok thank you, I was a bit overwhelmed by this question as they had given extra info that wasn't used in the calculation so I ought to think I was doing something wrong. I see my error here now. Thanks again!

haruspex · Mar 7, 2020

Bolter said:

they had given extra info that wasn't used in the calculation

That ought to be a more common practice. It is what happens in the real world. Deciding what information is relevant is the first challenge.

Calculate the velocity when 2 progressive waves are added together

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