1-5 Speed of a sinusoidal traveling wave

When a transverse sinusoidal wave propagates to the right along a string, the displacement of an element of the string located at position x can be described by

This element moves in a direction which is transverse to the motion of the wave.  The velocity associated with this transverse motion is the rate at which the element is moving up and down and it is called the transverse velocity of the element. To distinguish it from the velocity of the wave, we use u as the symbol for the transverse velocity.

The transverse velocity u of an element changes in a sinusoidal manner as a function of time and position while the velocity of the wave v is a constant quantity.

The transverse speed is maximum when the displacement of the element is zero and the transverse speed is zero when the displacement is maximum or minimum.