Trasverse wave wavelenght and velocity problem

[mmoadi]

Homework Statement

Along a long tense rope are traveling transverse waves with amplitude A= 10 cm. What is the wavelength, if at any given time the distance between two consecutive neighboring waves y= 5 cm is d= 6 cm? With what velocity are the waves spreading, if the maximum transverse speed of the rope is ω= 0.1m/s?

Homework Equations

v= fλ
k= 2π/ λ
y= A sin(k*x)

The Attempt at a Solution

For the wavelength:
k= 2π/ λ → λ= 2π/ k

Calculating k:
y= A(sin k)(sin x) → sin k= y/ A(sin x)
sin k= 5, which is wrong

Where is my mistake?

 

[ideasrule]

I don't understand what you did, but the equation for a wave is y=Asin(wt). (Note that the w here is different from the w in the question.) Derive that with respect to t and you get dy/dt=Awcos(wt), so Aw is the rope's maximum transverse speed. From there you can get the frequency, which you can then multiply by the wavelength to get speed.

 

[kerimek]

Your mistake is in assuming that the value of y is equal to the amplitude A. In this problem, y represents the distance between two consecutive waves, not the amplitude. Therefore, the correct equation to use is:

k= 2π/ λ → λ= 2π/ k

Calculating k:
y= A(sin k)(sin x) → sin k= y/ (A sin x)
sin k= 6/ (10 sin x)

Now, to solve for the wavelength, we need to find the value of sin x. This can be done by using the maximum transverse speed ω and the frequency f, which is equal to the number of waves passing a given point in one second. Therefore, we can use the formula v= fλ to find the frequency:

v= fλ → f= v/ λ
f= (0.1 m/s)/ (6 cm)
f= 0.01 Hz

Now, we can use the formula v= fλ to solve for the wavelength:

v= fλ → λ= v/ f
λ= (0.1 m/s)/ (0.01 Hz)
λ= 10 m

Therefore, the wavelength is 10 m. To find the velocity, we can use the same formula and substitute the frequency we just calculated:

v= fλ → v= (0.01 Hz)(10 m)
v= 0.1 m/s

The velocity at which the waves are spreading is 0.1 m/s.