Standing waves and harmonic waves

[glasshut137]

[SOLVED] standing waves

Homework Statement

A standing wave is a superposition of two harmonic waves described by
y1= Asin(kx+wt) and
y2= A sin(kx-wt),

where A=3.31594 cm, k=10.0531 m-1, and w=18.2212 s-1.

Determine the smallest positive value of x correspoding to a node.

So i added the two waves together and got y=2Asin(kx)cos(wt). I set y=0 and tried solving for x but i just got zero which was incorrect. Can someone help me solve these types of problems. Thanks.

 

[olgranpappy]

there are a lot of x values for which

0=2Asin(kx)

is true... x=0 is one of them.

to find the other zeros you have to know the zeros of the sin function... i.e., 0,pi,2pi,3pi, etc

so set kx equal to the smallest *nonzero* value listed above to find the node you want.

 

[Moetasim]

I can provide some insight on standing waves and how to solve problems related to them. A standing wave is a type of wave that occurs when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a wave pattern that appears to be standing still, hence the name "standing wave." In the context of the problem given, the two waves described by y1 and y2 are traveling in opposite directions and interfering with each other to form a standing wave.

To determine the smallest positive value of x corresponding to a node, we need to understand the concept of nodes and antinodes in standing waves. A node is a point in a standing wave where the amplitude of the wave is always zero, while an antinode is a point where the amplitude is always maximum. In the case of a standing wave, there will be nodes and antinodes that alternate with each other.

To solve the problem, we need to find the condition where the two waves y1 and y2 interfere to create a node. This can be done by setting y1+y2=0 and solving for x. We can rewrite the equation as 2Asin(kx)cos(wt)=0. Since the cosine function can only be zero when the angle is 90 degrees or multiples of 90 degrees, we can set cos(wt)=0 and solve for x. This will give us the condition for a node, where the two waves completely cancel each other out.

Plugging in the values given in the problem, we get cos(wt)=0 when wt=90 degrees or 270 degrees. This means that the smallest positive value of x corresponding to a node will be when wt=90 degrees, or when t=0.0049 seconds. We can then use this value of t to solve for x by setting cos(wt)=0 and solving for x. This will give us x=0.031 meters, which is the smallest positive value of x corresponding to a node in this standing wave.

I hope this explanation helps you solve similar problems in the future. Remember to always understand the concept and apply it systematically to solve the problem.