Finding the magnitude of the nearest peak and period

[apbuiii]

Homework Statement

Harmonic wave is described by: h(x,t)=cos(2∏(2x-t/2)). The equation is expressed as the height of the wave

At time t₀ a peak of the wave is at position X₀. What is the magnitude of the distance to the nearest peak at this time? How much time passes before another peak is observed at that postion?

Homework Equations

Period=2∏/B; Acos(B(X-D))

The Attempt at a Solution

I figured that the first part of the question is asking the wavelength and the second part of the question is finding the period. I simplified the equaiton to cos(4∏(X-t/4)). I don't really know how to distinguish the wavelength from the period from the given equation. They both make sense to equal 1/2.

Thanks for your help in advance!

 

[Simon Bridge]

I figured that the first part of the question is asking the wavelength and the second part of the question is finding the period.

That would be correct :)

Consider the general case: some pulse has a height function f(x) at t=0 - we write: h(x,0)=f(x) ... no brainer really. But if it moves to the right at speed v, then at time t it has moved a distance vt and we have to write:
h(x,t)=f(x-vt)

This is key to what you want to know!

Imagine f(x)=Acos(kx) ...this is your traveling wave at t=0, so at t>0 ... you can take it from here :)

 

[apbuiii]

Ah, I see what you're saying and I am taking what you left there for me :) Haha. I guess the introduction of two variables at a time confused me haha. One thing at a time... Thank you!

 

[Simon Bridge]

Good - since you took the trouble to simplify your equation, once you have expanded the general equation for h(x,t) you can just read-off k and v and use them to get the values you need.

 

[asteriatic]

I would approach this problem by first defining the terms "wavelength" and "period". Wavelength refers to the distance between two consecutive peaks or troughs of a wave, while period refers to the time it takes for one complete cycle of the wave. In this case, the given equation h(x,t) represents a harmonic wave with a frequency of 2∏ and a phase shift of t/2.

To find the magnitude of the distance to the nearest peak at time t0, we can substitute t0 into the equation and solve for x. This will give us the position of the peak at time t0. To find the distance to the nearest peak, we can simply take the absolute value of the difference between x0 and the position of the nearest peak. This will give us the magnitude of the distance.

As for the period, we can use the equation Period=2∏/B, where B represents the frequency. In this case, B=2∏, therefore the period is 1 second. This means that it takes 1 second for the wave to complete one cycle and for another peak to be observed at the same position.

In conclusion, the magnitude of the distance to the nearest peak at time t0 is 1/4 and it takes 1 second for another peak to be observed at the same position.