Why the velocity of a wave only depends on its medium? I can't figure out why it is constant in one kind of medium even when frequency or wavelength changes.
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1$\begingroup$ This isn't the case for light: the index of refraction varies by wavelength; and is n=c/v for that media. This accounts for the dispersion of light. The effect is negligible in air and non-existant in vacuum. $\endgroup$– Peter DiehrCommented May 28, 2018 at 6:02
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$\begingroup$ Assuming monochromatic light beams, then the only factors that effect the speed are the medium's permittivity and permeability since $c=\frac1{\sqrt{\varepsilon\mu}}$. With regards to different wavelength you would need to consider dispersion relations and group velocities $\endgroup$– JepsilonCommented May 28, 2018 at 7:56
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$\begingroup$ The speed of a wave in water (or any fluid) also depends on the wavelength. $\endgroup$– D. HalseyCommented May 28, 2018 at 22:52
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2 Answers
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The other answer provides a great analogy but I would like to add that waves are essentially propagation of energy through oscillations. Now the speed of a wave remains constant as you can think that the initial disturbance to create the wave was caused by a certain amount of energy and just like certain springs have only certain extensions when given a particular energy($E=1/2kx^2$) where $k$ is the spring constant and $x$ is the extension. So as this sort of spring constant, or shear modulus suppose for sound transmission through solids or compressibility and density etc. cause the energy dissipation to follow a certain pathway. This constant actually arises from the spacing between atoms in the medium and intermolecular forces on how they can vibrate to transfer energy along the medium.
The speed of sound depends primarily on the properties of the medium: density (atomic weight), modulus (for solids), and adiabatic index (for gases). This means that it changes with composition of air (which is one reason why you sound funny when you speak after inhaling helium) and for a given gas, with temperature
Mathematically, for small amplitude sound in gas the speed is given by
$$c=\sqrt{\frac{K_s }{\rho}}$$ Where $K_s$ is Bulk modulus suppose here for solids.
Notice that light does not need a medium to travel through and its speed is constant for different reasons including how it’s just a postulate on how the universe works and based on great scientific evidence and solutions to electromagnetism.
answered May 28, 2018 at 7:03
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How easy a spring can be compressed and extended depends on the stiffness of the spring right? For simplicity, assume sound waves. It propagate through a medium with compression and rarefaction. This compression and rarefaction of a wave depend on the medium and hence the velocity of the wave depends on the medium only. The frequency and wavelength depends on the relative motion of source and the observer (relativity).
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$\begingroup$ Yes, but WHY does transverse motion of the string for example play no role in the speed of the sound, even though it is what causes the disturbance? Different amplitudes of the same frequency will clearly demand different transverse speeds (same fq and bigger amplitude = bigger distance covered in the same period)... $\endgroup$– DimitriCommented Sep 20, 2022 at 13:59