In the case of sound waves, the propagation speed is the same for both longitudinal and transverse waves. The speed of sound in a medium depends primarily on the properties of that medium, such as its density, elasticity, and temperature. It is not directly affected by the type of wave.
However, it's important to note that longitudinal and transverse waves have different characteristics in terms of their motion and oscillation directions.
In a longitudinal wave, the particles of the medium vibrate parallel to the direction of wave propagation. Think of compressing and expanding a spring in one direction—the disturbance is transferred along the spring in the same direction. Sound waves in air, for example, are longitudinal waves. The particles of air oscillate back and forth in the same direction as the sound wave travels.
On the other hand, in a transverse wave, the particles of the medium vibrate perpendicular to the direction of wave propagation. A common example of a transverse wave is a wave on a string. When you pluck a guitar string, for instance, the disturbance causes the particles of the string to move up and down while the wave propagates horizontally.
So, while the propagation speed of sound is the same for both longitudinal and transverse waves, their differences lie in the direction of particle motion and oscillation relative to the direction of wave propagation.