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How can we determine the velocity and frequency of a wave by only seeing it's equation. For example :
If Given,

$$y=A\cos(k_1x-w_1t)\sin(k_2x-w_2t)$$

an equation of a particular wave. Then what will be its velocity and frequency? Is there any generalised method for finding velocity and frequency from any given wave equation? Please explain.

QuantumOscillator
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MD Hossain
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2 Answers2

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Your function can be rewritten as $$ y = \frac{A}{2} \left[ \sin \big( (k_1 + k_2) x - (\omega_1 + \omega_2) \big) t - \sin \big( (k_1 - k_2) x - (\omega_1 - \omega_2) t \big) \right] $$ In this form, you can see that it's actually the superposition of two traveling waves, one with frequency $(\omega_1 + \omega_2)$ and (phase) velocity $$ v_a = \frac{\omega_1 + \omega_2}{k_1 + k_2} $$ and the other with frequency $|\omega_1 - \omega_2|$ and (phase) velocity $$ v_b = \frac{\omega_1 - \omega_2}{k_1 - k_2}. $$

Note that for many waves, the phase velocity of the wave is independent of the frequency. Such waves are called non-dispersive. If that's the case here, then it must be the case that $v_a = v_b$. After some algebra, we find that $v_a = v_b = \omega_1/k_1 = \omega_2/k_2$ in this case.

There are also situations in which we do not necessarily have $v_a = v_b$; such waves are called dispersive. If you're just learning about waves and that word doesn't mean anything to you, then don't worry about it; non-dispersive waves are far more common, particularly in introductory physics classes.

Finally, you ask whether there is any way to find the generalised velocity and frequency of a wave function. The answer is yes; it's called the Fourier transform of the function. Effectively, any function of $x$ and $t$ can be thought of as a superposition of (an infinite number of) travelling waves, each with its own wave number $k$ and frequency $\omega$. The phase velocity for each one of these components would then just be $\omega/k$. A detailed description of how this technique is performed would be pretty involved, and I can't possible do justice to it here. If you're interested in learning more about this topic, I would recommend that you start with the simpler idea of a Fourier series; once that makes sense to you, start reading up on Fourier transforms.

Michael Seifert
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  • The what will be the Velocity of my wave? w1/k1=w2/k2 can we evaluate the wave velocity from this equation? – MD Hossain Sep 21 '21 at 10:51
  • @MDHossain The speed is equal to $\omega_1/k_1 = \omega_2/k_2$. I've made an edit to my answer to make this clearer. – Michael Seifert Sep 21 '21 at 11:26
  • How can we calculate phase and group velocity from the above equation?? – MD Hossain Sep 21 '21 at 16:02
  • So if $\omega_1 k_2≠\omega_2 k_1$ and if i am not considering $k_1$~$k_2$ then this wave will not move in a constant velocity, that i learned so far. \ am i right? – MD Hossain Sep 23 '21 at 19:45
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This is not a simple wave, and it doesn't have a single velocity. It is a product of two waves each of which has a different velocity (unless $\omega_1k_2=\omega_2k_1$). Can you refine the question?

CWPP
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  • What are the velocities if w1k1≠w2k2? Are you saying about group velocity and phase velocity? – MD Hossain Sep 21 '21 at 11:05
  • What happens if $v_a$≠$v_b$? Will then the resultant wave move in a constant velocity or not? – MD Hossain Sep 21 '21 at 13:29
  • The wave does not propagate with a single velocity. If you know about phase and group velocity then you will know that the zeros and peaks of the wave travel at the phase velocity and the beats travel at the group velocity. See (https://users.physics.ox.ac.uk/~palmerc/Wavesfiles/dispersive_wavepacket.htm) for a visualisation – CWPP Sep 21 '21 at 15:51
  • Yeah i know about phase and group velocity.. But when we derive phase and group velocity we assume $k_1$~$k_2$ and $w_1$~$w_2$. What if they differ much from each other? – MD Hossain Sep 21 '21 at 16:00
  • So if $\omega_1 k_2≠\omega_2 k_1$ and if i am not considering $k_1$~$k_2$ then this wave will not move in a constant velocity, that i learned so far. \ am i right? – MD Hossain Sep 23 '21 at 19:43
  • The wave shape will change with time and so you cannot assign a single velocity to it. Do check out one of the many available visualisations, such as mine in the previous comment. – CWPP Sep 25 '21 at 13:02