Wave on a rope - question concerns the maths of the wave equation

[karnten07]

Homework Statement

[(w^2).b - Tk^2]/Qw = tan(kx - wt + P)

This can't be solved for all (x,t) with constant values of w and k

Can you explain why this is so please?

ive used b to represent the mass per unit length, and T is the tension

Homework Equations

This is the answer to a question that asked if why a particular value of y doesn't satisfy the general wave equation. I just don't understand why the statement is true.

The Attempt at a Solution

i think this may just be a mathematical explanation due to the nature of the tan function, but I am unsure.

 

[Kurdt]

I think you're on the right lines. Have you looked at the tan function to see what its nature is?

 

[Astronuc]

Looking at

[(w^2).b - Tk^2]/Qw = tan(kx - wt + P)

let w, k, Q and T be constants, and K = [(w^2).b - Tk^2]/Qw

then K = tan (kx - wt + P),

tan^-1(K) - P = C = kx - wt and C = kx - wt is the equation of a line, which then means that there is only one x for one t.

On the other hand, T would be a function of lateral displacement.

 

[karnten07]

I should clarify that the motion of the rope that is being studied is the vertical motion as a wave travels along a horizontal lying rope, so in the direction y. The wave equation that is used is

b. (d^2y/dt^2) = T.(d^2y/dx^2) - Q.dy/dt

The derivative multiplied by Q is the part of the equation that describes the fricitional forces of the rope that are proportional and opposite to the velocity of the rope.

The y that is used is a cosine function and when inserted into the equation provides the end formula as stated in the first post.

So from this, i think that the tension will vary along the rope due to the differing motion of parts of the rope. Is this what you mean by it bein a function of lateral displacement?

Im not sure i understand the statement that there would be one value of x for a value of t, because at a time t, the rope would have just one value of x for a time t unless the rope had coiled back on itself above x, therefore giving two values. But this motion is considered not to happen here, so it would be true that one value of x occurs for one value of t.

Any clarification on how the formula arrived at by substituting our value of y, shows that the value of y isn't a vlaid solution to the wave equation given?