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…
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.
View the full transcript and gain access to JoVE Core videos
Q1: What factors determine the speed of a transverse wave on a string?
The speed of a transverse wave depends on two physical quantities: the tension in the string and its linear mass density, defined as mass per unit length. Increasing tension increases the restoring forces that straighten the string when disturbed, thus increasing wave speed. Conversely, increasing mass makes motion more sluggish and decreases speed.
Q2: How does tension affect wave propagation on a rope?
Tension forces in the rope are tangent to its surface at each point. The tension components in the x-direction cancel out, while the y-components determine the net restoring force. This restoring force accelerates the rope element perpendicular to wave motion, directly influencing how quickly the wave travels through the medium.
Q3: Why do guitar strings with different thicknesses produce different frequencies?
Guitar strings have different linear densities based on their thickness, even when made from similar material. Since wave speed depends on both tension and linear density, strings with different mass per length vibrate at different speeds. The wave speed and wavelength together determine the frequency of sound produced.
Q4: What is the mathematical relationship between wave speed, tension, and linear mass density?
The wave speed equation is derived from Newton's second law applied to a rope element. By analyzing the slope of the rope at different points and combining force expressions, the wave speed formula emerges in the same form as the linear wave equation, showing speed as a function of tension divided by linear density.
Q5: How does a rope element move when a transverse wave passes through it?
When a transverse wave travels along a rope, individual rope elements oscillate perpendicular to the wave's direction of motion. This perpendicular oscillation is caused by the rope's restoring force, which acts to return the element to equilibrium. The element does not travel with the wave; only the wave pattern propagates forward.
Q6: Why is light from lightning seen before thunder is heard?
Light waves propagate much faster than sound waves in air. Because the speed of transverse waves depends on medium properties, electromagnetic waves travel at approximately 3 × 10^8 m/s, while sound travels at roughly 343 m/s. This enormous difference in wave speed explains why we observe the flash before hearing the sound.
Q7: Why is understanding transverse wave speed important for musical instruments?
Wave speed on strings is essential for analyzing stringed musical instruments because it directly affects the frequencies produced. Additionally, the mathematical expression for wave speed on strings applies to many kinds of mechanical waves, making it a fundamental principle in wave physics and acoustics.