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I've seen a mathematical example where a wave was made from same source in water and air. but in both medium frequency was same but wavelengths were different in each medium. we know that velocity of a wave = frequency * wavelength. if the velocity of a wave changes from medium to medium(wave;in two different medium but made from same source) why doesn't the frequency change (in case of a wave which is made from same source)?

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    $\begingroup$ possible duplicate of Why frequency doesn't change during refraction? $\endgroup$ – John Rennie Nov 10 '14 at 12:42
  • $\begingroup$ It doesn't seem duplicate since this questions doesn't ask about a wave that travels from one medium to a different one (refraction) but rather about having the same source in two different mediums. $\endgroup$ – Asaf Nov 10 '14 at 13:32
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    $\begingroup$ yes, i have not asked about refraction, in fact i have not read about wave refraction yet !! $\endgroup$ – tahsin Nov 10 '14 at 14:37
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It is best to first simplify the situation and think about why is the frequency of the wave the same as the source.

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Considering the case of a string attached to an oscillator, if the frequency of the oscillator and the wave were different, there would need to be a discontinuity in the string. The fact that the string is attached to the oscillator means they need to oscillate at the same frequency.

Now, in your case you describe a situation where the same source is attached to two different mediums. This is analogous to an oscillator attached to two strings of different thickness. As shown, the frequency must be the same. The speed of propagation depends on the medium. In the case of a string it will be $$ v = \sqrt{\frac{\text{tension}}{\text{mass per unit length}}}. $$ Then, the wavelength must change accordingly to keep the $v/\lambda$ ratio constant.

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