16.6:
Speed of a Transverse Wave
Consider a flexible taut rope in an equilibrium position with a linear mass density μ and a tension force T.
When a constant upward force is applied at the right end, a transverse wave travels through the rope with constant speed.
The small element of length Δx oscillates perpendicular to the wave motion because of the rope's restoring force.
The force at each end is tangent to the rope.
The tension forces in the x-directions have equal magnitude and opposite directions, so they cancel.
The slope of the rope at points x and x plus Δx determines the expressions for the y-components of the force.
Combining these expressions gives the net y-component of the force, which according to Newton's second law, equals the elements' mass times the y-component of acceleration.
Dividing by T-Δx and taking the limit Δx to be zero gives the expression, in the same form as the linear wave equation.
This provides the expression for the speed of the wave, which depends on the tension and the linear density.
16.6:
Speed of a Transverse Wave
The speed of a wave depends on the characteristics of the medium. For example, in the case of a guitar, the strings vibrate to produce the sound. The speed of the waves on the strings and the wavelength determine the frequency of the sound produced. The strings on a guitar have different thicknesses but may be made of similar material. They have different linear densities, and the linear density is defined as the mass per length.
One of the key properties of any wave is the wave speed. Light waves have a much greater propagation speed than sound waves in the air. For that reason, a flash is seen from a lightning bolt before the clap of thunder is heard. It is important to understand the speed of transverse waves on a string because this speed is essential for analyzing stringed musical instruments. Furthermore, the speeds of many kinds of mechanical waves have the same primary mathematical expression as the speed of waves on a string. The physical quantities that determine the speed of transverse waves on a string are the tension in the string and its mass per unit length (also called the linear mass density). Increasing the tension increases the restoring forces that tend to straighten the string when disturbed, thus increasing the wave speed. Increasing the mass makes the motion more sluggish and decreases the speed.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson; section 15.4; pages 482–485.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/16-3-wave-speed-on-a-stretched-string section 16.3; pages 805-807.