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An ultrasonic wave is sent through a dispersive medium. That means the waves move at different frequencies with different phase velocities. If this were measured in the frequency spectrum, would the dispersion (or wave speed) have an influence on the amplitude? Is there a connection?

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Strictly speaking, dispersion does not affect the magnitude of the spectrum, it only affects the phase (associated with the phase speed). However, if you have a non-trivially dispersive medium it is often the case that you will also have absorption of some sort (in fact, this is required by causality through the Kramers-Kronig relations), and that absorption will affect the magnitude.

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