- #1
Saitama
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Homework Statement
I am posting three problems together as I believe they are quite short.
1.Show that the particle speed can never equal to the wave speed in a sine wave if the amplitude is less than wavelength divided by ##2\pi##.
2.Two wave pulses identical in shape but inverted with respect to each other are produced at the two ends of a stretched string. At the instant when the pulses reach the middle, the string becomes completely straight. What happens to the energy of two pulses?
3.Show that for a wave traveling on a string
$$\frac{y_{max}}{v_{max}}=\frac{v_{max}}{a_{max}}$$
where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to write $$\frac{y_{max}+v_{max}}{y_{max}-v_{max}}=\frac{v_{max}+a_{max}}{v_{max}-a_{max}}$$
(By usual notation, y is the displacement, v is transverse velocity and a is the acceleration of any particle on the string)
Homework Equations
The Attempt at a Solution
Attempt for 1.
Let the string wave be represented by the equation ##y=A\sin(\omega t-kx)##. The transverse velocity of particle at position at any instant of time t is given by ##∂y/∂t=A\omega \cos(\omega t-kx)##. The speed is hence given by:
$$\sqrt{A^2\omega^2\cos^2(\omega t-kx)+\frac{\omega^2\lambda^2}{4\pi^2}}$$
where the second term is the wave speed given by ##\omega/k=\omega\lambda/(2\pi)## and ##\lambda## is the wavelength. Since ##A \leqslant \lambda/(2\pi)##, we have
$$\sqrt{A^2\omega^2\cos^2(\omega t-kx)+\frac{\omega^2\lambda^2}{4\pi^2}}\leqslant \frac{\omega \lambda}{2\pi}\sqrt{1+\cos^2(\omega t-kx)}\leqslant \frac{\sqrt{2}\omega \lambda}{2\pi}$$
The above does not agree with the problem.
Attempt for 2.
I honestly have no idea about this one. :(
Attempt for 3.
It is very easy to find the relations presented in the problem statement by assuming the wave equation to be ##y=A\sin(\omega t-kx)##. I am not sure why the author asks the second part of the problem. I don't see why it would be wrong to do some simple algebra on the relation obtained.
Any help is appreciated. Thanks!