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If I am emitting an ultrasonic wave through a material and collecting that wave at the end of my material (in a pitch-catch setup), would my frequency response change if my wave velocity changed?

For instance, if I emit an ultrasonic wave at 100 kHz with a wave velocity of 8000 m/s, collected the signal at the other end of my material and computed the Fast Fourier Transformation, I expect to get a peak at 100 kHz from the FFT (when looking at Amplitude vs Frequency).

Would the FFT response change if I emitted the same wave at 100 kHz, on the same material but this time managed to send the wave at 5000 m/s (by changing the temperature of the material)?

For a more practical example, if I look at a figure from: (https://doi.org/10.1177/1475921710365267), they show a change in the signal obtained due a different wave velocity. Would this cause the FFT frequency response to be the same or different? The paper goes onto say the testing setup is identical, simply the temperature is increased which will change the sound velocity.

Figure 2 from https://doi.org/10.1177/1475921710365267

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The behavior of a wave in a medium is governed by the dispersive properties of that medium--i.e. the dispersion relation. The dispersion relation provides a relationship between the frequency, $\omega$ and wavenumber $k$ of a wave. Given the dispersion relation, you can compute the group velocity and phase velocity of a wave.

You can't exactly choose the velocity of your sound wave with the equipment that produces it, but you can pick the frequency. The dispersion relation of the material will then determine the phase and group velocities of the wave that propagates through it at that frequency.

For example, the dispersion relation for electromagnetic waves in a vacuum is $$\omega = c k,$$ and thus the phase velocity is constant at all frequencies: $$v_p = \frac{\omega}{k} = c.$$ Whereas, deep water waves obey a dispersion relation: $$\omega = \sqrt{g k},$$ which admits a frequency-dependent phase velocity.

Thus, the notion of choosing the velocity of a wave independent of its frequency make no sense. The velocity is generally determined by the frequency.

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  • $\begingroup$ I had read a paper which showed two signals, each used with the same apparatus and material, however conducted at different temperatures. In the paper, they show the signals to have different amplitudes and arrival times, which can be attested to the sound velocity changing. If they had computed the FFT on both signals would they obtain an identical response ? (doi.org/10.1177/1475921710365267) $\endgroup$
    – iato
    Commented Feb 16, 2023 at 6:06
  • $\begingroup$ I edited my original question to show the Figure from the paper. $\endgroup$
    – iato
    Commented Feb 16, 2023 at 6:09
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    $\begingroup$ It makes sense that changing temperature could alter the dispersive properties of a material. However, the change in velocity is an effect of the temperature change for a given applied frequency. If you put in a wave packet rather than a wave at a single frequency, the dispersive properties of the material would cause the Fourier components of the packet to travel at different speeds and thus cause the packet to spread out. I assume that if you change the temperature, you will change the dispersive properties and thus change the degree to which the wave packet spreads as it propagates. $\endgroup$
    – klippo
    Commented Feb 16, 2023 at 6:22
  • $\begingroup$ Would the change in how the packet spread alter the frequencies itself as it propagates or simply the amplitude at 100 kHz ? $\endgroup$
    – iato
    Commented Feb 16, 2023 at 6:26
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    $\begingroup$ If a material is nonlinear and/or you have large perturbations, yes the dispersion relation can become complicated and involve the amplitude of your wave. This will cause attenuation of the wave. However, the frequency will not be changed. $\endgroup$
    – klippo
    Commented Feb 16, 2023 at 6:35

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