Question

How is the wave speed of a string the same as calculating the speed using kinematics?

Answer 1

Waves oscillate in a string as transverse waves. Waves have different properties which are used to describe them like wavelength, amplitude, frequency etc. The wave speed is related to wavelength and frequency while speed in kinematics is related to distance and time.

Complete solution:
A wave is a disturbance which moves forward by transferring its disturbance to the other molecules in contact. It has different properties like wavelength, frequency, speed etc.
The distance over which the shape of a wave repeats is known as wavelength. In a transverse wave, it can be defined as the distance between two troughs or crests. Its SI unit is metre ($m$).

Frequency is the number of oscillations of a wave that occurred in one second. It is the reciprocal of time period. Its SI unit is hertz ($Hz$).

The time in which one oscillation is completed by a wave is known as the time period of the
wave. Its SI unit is seconds ($s$).

Speed of the wavelength is the distance travelled by a point on a wave in unit time. Its SI unit is $m{{s}^{-1}}$.

The relation between wavelength, frequency and speed is given by-
$v=\lambda \nu$
Here, $v$ is the speed
$\lambda$ is the wavelength
$\nu$ is the frequency

We know that,
$\nu =\dfrac{1}{T}$
Here, $T$ is the time period

Substituting in above equation, we get,
$v=\dfrac{\lambda }{T}$ - (1)
Speed in kinematics is given by-
$s'=\dfrac{d}{t}$ - (2)
Here, $d$ is the distance
$t$ is the time taken

Comparing eq (1) and eq (2), we get,
Wavelength is the distance travelled by a point on the wave and time period is the time taken to travel by the point on the wave to travel the distance. Hence, we can say that the speed of a wave is similar to the speed calculated in kinematics.

Therefore, from eq (1) and eq (2), the wave speed of a string is the same as speed in kinematics.

Note:
The wave speed of a string can be assumed as the speed in kinematic of a point moving through the length of the string. A string produces transverse waves which move by forming crests and troughs. Electromagnetic waves are transverse waves; they do not require a medium to propagate.