Wave speed Knowledge-Based Study

Study notes

1. What is Wave Speed?

   Wave speed refers to the rate at which a wave disturbance propagates through a medium. It is essential to distinguish between the wave speed (the speed at which energy and the wave pattern travel through space) and the particle speed (the speed at which individual particles of the medium oscillate about their equilibrium positions).

   For a periodic mechanical wave, one complete cycle of oscillation produces exactly one wavelength of the wave pattern. If the period of the oscillation is denoted by T and the wavelength by $\lambda$, the wave travels a distance of $\lambda$ in a time interval equal to T.

   Because frequency is the reciprocal of the period ($f = \frac{1}{T}$), the fundamental wave relationship is expressed as:

   $v = \frac{\lambda}{T} = f\lambda$

   • $v$: wave speed, measured in meters per second ($\text{m/s}$)
   • $f$: frequency of the source, measured in hertz ($\text{Hz}$)
   • $\lambda$: wavelength, measured in meters ($\text{m}$)

   How the Medium Dictates Wave Speed

   A crucial principle in wave mechanics is that the speed of a mechanical wave is determined primarily by the physical properties of the medium through which it travels, not by the frequency or amplitude set by the wave source.

   In general, the speed of a mechanical wave depends on a balance between two main physical attributes:

   • Elastic properties: The tendency of the medium to return to its original shape when deformed. Greater stiffness or tension results in faster wave propagation.
   • Inertial properties: The mass density or linear density of the medium. Greater mass or inertia resists acceleration, resulting in slower wave propagation.

   For a transverse wave propagating along a stretched flexible string, this relationship is given quantitatively by:

   $v = \sqrt{\frac{F_T}{\mu}}$

   where FT represents the tension in the string in newtons ($\text{N}$) and $\mu$ represents the linear mass density in kilograms per meter ($\text{kg/m}$). Increasing tension increases the speed, whereas using a heavier string decreases the speed.

   Applying the Principles

   Understanding how wave variables interact when conditions change is vital for solving problems in wave mechanics:

   1. Changing Frequency in a Uniform Medium

   If the source frequency f is doubled while a wave travels within the same unaltered medium, the wave speed v does not change. Instead, the wavelength $\lambda$ is halved according to $\lambda = \frac{v}{f}$.

   2. Crossing a Boundary into a Different Medium

   When a mechanical wave travels from one medium into another (for example, from a light string to a heavy string):

   • The frequency ($f$) remains strictly constant because it is governed entirely by the driving source at the boundary.
   • The speed ($v$) changes due to the new mechanical properties of the second medium.
   • The wavelength ($\lambda$) changes proportionally with the speed to satisfy $v = f\lambda$.

   Thus, an increase in propagation speed across a boundary causes an increase in wavelength, while the frequency remains unaffected.