What Is the Distance Between Adjacent Nodes in Superimposed Waves?

[vivekfan]

Homework Statement

1)Two Traveling Waves y1=Asin[k(x-ct)] and y2=Asin[k(x-ct)] are superimposed on the same string. What is the distance between adjacent nodes?

2) Standing Waves are produced by the interference of two traveling sinusoidal waves, each of frequency 100 Hz. The distance from the second node to the fifth node is 60 cm. What is the wavelength of the original waves?

Homework Equations

y(x,t)= ymax sin (kx-wt), where w=omega

For the combined wave:

y(x,t)=[2ymax sin kx]cos wt, where w=omega

The Attempt at a Solution

For the first question, I read in my book that adjacent nodes are separated by half a wavelength, but I'm not sure why this is true, and also what is the wavelength in the combined wave in this question? Please help.

For the second question, I know that you find the nodes by equating sin kx to zero, but I'm confused about incorporating the distances?

 

[Easty]

Homework Statement

1)Two Traveling Waves y1=Asin[k(x-ct)] and y2=Asin[k(x-ct)] are superimposed on the same string. What is the distance between adjacent nodes?

2) Standing Waves are produced by the interference of two traveling sinusoidal waves, each of frequency 100 Hz. The distance from the second node to the fifth node is 60 cm. What is the wavelength of the original waves?

I think you ment y1=Asin[k(x-ct)] and y2=Asin[k(x+ct)] because these 2 wave will produce a standing wave once superimposed and this standing wave will have nodes at every half wavelengh. So a node is defined as a fixed point when the amplitude of the wave is 0. Its easy to see if you think about a graph of y=sin(x), This graph will look like the standing wave at any instant in time. So y=0 at x=0, $$\pi$$/2 and $$\pi$$, the difference between each point is half a wavelenght.

 

[vivekfan]

I think you ment y1=Asin[k(x-ct)] and y2=Asin[k(x+ct)] because these 2 wave will produce a standing wave once superimposed and this standing wave will have nodes at every half wavelengh. So a node is defined as a fixed point when the amplitude of the wave is 0. Its easy to see if you think about a graph of y=sin(x), This graph will look like the standing wave at any instant in time. So y=0 at x=0, $$\pi$$/2 and $$\pi$$, the difference between each point is half a wavelenght.

How do you know what the wavelength of the resulting wave is?

 

[Easty]

The wavelenght of each inital wave is the same, so once superimposed the new wave will have the same wavelenght, which will be given by ( lambda= 2*pi/k)

So each node will be located at Lambda=pi/k