Unit 1.1 Speed of Waves
Speed of Longitudinal Waves
— The speed of a wave depends on the properties of the medium in which it travels.
— The speed of a longitudinal wave on a stretched string or cord depends on the tension in the cord and on the mass per unit length of the cord.
Example: A wave whose wavelength is 0.30 m is traveling down a 300m long wire whose total mass is 15 kg. If the wire is under a tension of 1000 N, what are the speed and frequency of this wave?
Use the formula where μ is the mass per unit length. Eq 1
Answer:
Velocity of this wave on a wire is given by f = ^{v}/_{λ}
Frequency = ^{140}/_{0.20}
Frequency = 470 Hz
Exercise 1.1.7
(a) What would be the effect of increasing the tension of the wire?
(b) What would be the effect of increasing the thickness of the wire?
(c) What would be the effect of increasing the density of the wire?
Speed of Transverse Waves
— The speed of a wave depends on the properties of the medium in which it travels.
— The speed of a transverse wave on a long solid rod depends on the ‘elastic force factor’ and its densitywhere μ is the mass per unit length.
— The speed of a longitudinal wave traveling in a stretched wire is where E is the elastic modulus and ρ is the density.
— The speed of a longitudinal wave traveling in a liquid or gas where B is the bulk modulus and ρ is the density.
Example:
(a) Estimate the wavelength of a dolphin’s echolocation wave. (b) If an obstacle is 100 m from the animal, how long after the animal emits a 100,000 Hz wave is its reflection detected? Density of sea water is 1.025X103kgm^{3} and the bulk modulus is 2.0X109Nm^{2}. You may use the equation for the speed of a longitudinal wave traveling in a liquid or gas, where B is the bulk modulus and ρ is the density.
Answer:
The speed of longitudinal waves in sea water:
Then:
The time is
The time required for the round trip between the animal and the object is
Question: Why do water waves ‘break’ near the shore?
— A shallow water is a combination of transverse and longitudinal wave motions.
— The motion of each particle of water at the surface is circular or elliptical.
— At the bottom, the motion is only horizontal.
— When a wave approaches shore, the water drags at the bottom and is slowed down, while the crests move ahead at higher speed.
— As a result the top ‘falls’ over the base.
Exercise 1.1.8
1. (a) Why are there breakers further out at sea when the wind is strong?
(b) Why are there ‘white caps’ in the deep ocean when the wind is strong?
2. A strong gust of wind creates a wave of wavelength 0.50 m on one end of a T&TEC 200mlong high tension wire. If the total mass of the wire is 20 kg and the wire is under a tension of 490N, what are the (a) speed and (b)
frequency of this wave? You may use the equation for the speed of a transverse wave on a long solid rod, where μ is the mass per unit length.
