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Profile is the right word, also response curve. If you can find one, try to look for a waterfall graph of “frequency, intensity, decay” to get a better idea of what to expect.
Studio headphones, or ones that you can most easily adjust the response with some EQ, have the flattest response curve and the shortest decay.
There is also transient response like with any electric circuit, but I haven’t seen anyone do a full analysis for headphones.
Thanks for these terms. One problem with google is how they always end up with big name domains and they basically use the same terms. And I need these terms you used so I can search in more depth. Thanks again.