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Is it possible for two wave to have same wavelength but different speed and frequency?

According to $C=λ×f$, if wavelength is constant, then frequency must be proportional to speed.

My explanation is : It is impossible for two waves to have different speed and frequency but same wavelength , because , Speed is distance covered per unit time . If speed increases then wavelength must be increase , therefore two waves can't have same wavelength while having different speed and frequencies Am I right? If not pls explain

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Sure. For an easily visualized example, if I have two identical cylinders, one filed with air and one filled with water then the lowest frequency standing sound waves in the two cylinders will have the same wavelength, set by then length of the cylinders, but one will have a speed set by the speed of sound in air and the other by the speed of sound in water (which is significantly higher), and so they will have correspondingly different frequencies.

For a less contrived, but more complex example, waves traveling in a regular lattice, such as a crystalline solid, will typically have several possible frequencies (and correspondingly speeds) at the same wavelength. These different possible frequencies are said to be in different 'bands' and plotting the different allowed frequencies as a function of wavelength (or more conventionally as a function of wavenumber $=\frac{2\pi}{\lambda}$) is said to give the band structure of the material. Different types of waves in the crytal will have different band structures, so sound waves will have a 'phonon band structure' whilst quantum mechanical electron waves will have an 'electronic band structure'.

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In general yes, it is possible. Since $\lambda=C/f$, any set of $C$ and $f$ with the same ratio will give the same wavelength. e.g. the set $C=1\,\rm{m/s}$ and $f=1\,\rm{s^{-1}}$ as well as the set $C=2\,\rm{m/s}$ and $f=2\,\rm{s^{-1}}$ both give a wavelength of $1\,\rm m$.

Now, if you are looking at waves in the same medium, then things get more complicated. If there is no dispersion, then this is impossible, since the wave speed will be determined solely by the medium and must be the same for all waves. Then there can only be one set of valid $C$, $\lambda$ and $f$. If the wave speed does have a dependence on wavelength $\lambda$, then you would have to see if $C(\lambda)/\lambda$ is a one-to-one function or not to determine if you can have multiple valid frequencies.

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  • $\begingroup$ Speed is distance covered per unit time . If speed increases then wavelength must be increase $\endgroup$
    – user324098
    Commented Jan 4, 2022 at 18:52
  • $\begingroup$ @VaibhavTiwari You are assuming a constant frequency in your statement. $\endgroup$
    – BioPhysicist
    Commented Jan 4, 2022 at 19:00
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The speed of a mechanical wave, such as sound (in some gas) or a vibrating string, is determined by the material condition of the medium. For sound, the temperature and molecular mass of the gas determine the speed of the wave, $C$. The particular source (vocal folds or a loudspeaker cone) will determine the frequency mixture of the compression wave, and the wavelength or waveform in the gas will conform to $$\lambda = \frac{C}{f}$$ for each frequency component ( see Fourier series)

For a string, the material density, diameter, and tension will determine the speed, and again, the source will determine the frequency (not discussing resonant frequencies in this answer). Again, the wavelength will obey the previous equation.

If you have two different media conditions (two layers of a gas at two different temperatures) one could have two different frequency sources which would produce equal wavelengths in those two media. If one has a single homogeneous medium, however, it's not possible to have a common wavelength with two different frequencies.

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  • $\begingroup$ Speed is distance covered per unit time . If speed increases then wavelength must be increase $\endgroup$
    – user324098
    Commented Jan 14, 2022 at 13:12
  • $\begingroup$ Pls explain me , if speed change wavelength must change , so if two waves have different speed then their wavelength must be different $\endgroup$
    – user324098
    Commented Jan 14, 2022 at 13:14
  • $\begingroup$ @VaibhavTiwari It depends on what how the frequencies of the waves compare to each other. Two waves with the same frequency will have different wavelengths if the speeds of the waves are different. $\endgroup$
    – Bill N
    Commented Jan 14, 2022 at 14:30
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One could envisage such a situation in dispersive media, i.e., the media where the frequency and the wave number are related via a non-linear dependence, $\omega(k)$. For example, plasma oscillations present many situations where the oscillations frequency is independent on wave length, i.e., we can have waves of different wavelength but same frequency, see here for the list of dispersion laws.

See also this thread: Why do wave equations produce single- or few- valued dispersion relations?

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