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... sound waves don't seem to noticeably cause warmth
Ultrasonic welding of thermoplastics causes local melting of the plastic due to absorption of vibration energy.
The perception of sound in any organism is limited to a certain range of frequencies. For humans, hearing is normally limited to frequencies between about 20 Hz and 20,000 Hz (20 kHz), although these limits are not definite. The upper limit generally decreases with age. Other species have a different range of hearing. For example, dogs can perceive vibrations higher than 20 kHz, but are deaf to anything below 40 Hz.
I suppose that part of the answer is to recognize that the increased motion designated as "heat" is a random motion, with all local particles moving fairly independently, while "sound" is a coherent motion with local groups of particles moving in the same direction to produce a series of pressure waves.