From the sound of music reaching our ears to the ripples in a pond, waves are present everywhere. Have you ever pondered, though, how quickly these waves travel?
What is a Travelling Wave?
An energy-transferring disturbance that travels through a medium without moving matter is called a traveling wave. We think tossing a stone into a pond that is motionless. Waves moving through the water cause the ripples to spread out. Light waves move across space, and sound waves move through air.
Speed of a Travelling Wave
Each medium particle moves up and down (or back and forth) in a traveling wave, but it does not follow the wave’s path. While the particles merely oscillate around their locations, the wave travels, carrying energy.
The motion of a fixed phase point on the wave is given by ks – ωt = constant.
As time t changes the position x of the fixed phase point must change so that the phase remain constant.
ks – ωt = k (x + delta x) – ω(t + delta t)
k delta x – ω delta t = 0
Taking delta x, delta t vanishingly small, this give dx / dt = ω / k = v
v = 2 phi v /2 phi / λ = λv = λ / T
This is general relation for all progressive waves.
Speed of transverse wave on stretched string: The wave velocity through a medium depends on its inertial and elastic properties. So the speed of transverse wave through a stretched string is determined by two factors:
Tension T in the string is a measure of elasticity in the string. Without tension no disturbance can propagate in the string. Dimension of T = [Force] = [MLT-2]
Mass per unit length or linear mass density of the string so that the string can store kinetic energy.
Dimensions of mass density = [Mass] / [Length] = [ML-1]
Now dimension of ratio T / mass density
[MLT-2] / [ML-1] = [L2T-2].
As speed v has the dimensions [LT-1], so we can express v in term of T and mass density as v2 = T / mass density.
Wave Speed
A travelling wave’s speed, which is simply the distance a point on the wave (such as a crest or a trough) travels in a specific amount of time, indicates how quickly the disturbance (the wave) moves through the medium.
Mathematically, the speed v of a wave is given by:
If we know the wavelength (λ) the distance between two consecutive crests (or troughs) and the frequency (f) how many waves pass a point in one second we can use a simple formula:
v = λ × f
This is one of the most important and easy-to-remember formulas in wave motion.
Breaking it Down: Wavelength and Frequency
Wavelength (λ): This is the length of one complete wave cycle. Think of it as the “size” of one wave.
Frequency (f): This is how often the wave cycles occur per second. It is measured in hertz (Hz).
Depending on the medium, a wave with a big wavelength and a low frequency may move slowly or swiftly. How wavelength, frequency, and speed relate to one another is essential to understand wave behaviour.
Factors Affecting the Speed of a Wave
A wave’s speed is mostly determined by the material it is passing through. The speed at which waves travel varies depending on the material. A few key points are as follows:
Waves often travel quicker in solids due to the particles’ close proximity and rapid energy transmission.
Waves travel more quickly in liquids than in solids but more slowly in gases due to the distance between the particles.
The speed of sound, for example, is roughly 343 m/s in air, 1500 m/s in water, and 5000 m/s in steel.
Wave speed can also be influenced by temperature, particularly in gases. Higher temperatures mean particles move faster, allowing the wave to travel quicker.
Example to Understand Better
To make waves, standing at one end of a long rope and flicking it up and down. How tight the rope is and what kind of material it is constructed of determine how quickly the wave moves along it.
The wave will move more quickly if we increase the strain in the rope. The wave may slow down if the rope is made of a heavier material.
Similar to this, tightening a string on a musical instrument like a guitar causes the wave to travel faster and alters the pitch of the sound we hear.
Summary
Travelling waves move through a medium carrying energy.
The speed of a travelling wave tells us how fast the disturbance moves.
Speed is related to wavelength and frequency by the formula: v = λ × f.
The medium and its properties (like tension, density, and temperature) greatly affect the wave’s speed.
The rate at which a traveling wave travels across a medium is known as its speed. It displays the speed at which the energy or disturbance of the wave is transferred from one location to another.
The speed v of a wave is given by the formula:
v = λ × f.
Where λ is the wavelength and f is the frequency of the wave.
A wave’s speed is mostly determined by the density and elasticity of the medium as well as, in certain situations (such as gases), temperature. In general, wave speed increases with temperature and tighter particles.
A wave’s speed in a given medium doesn’t change even if its frequency does. To maintain a consistent speed, the wavelength adapts to changes in frequency.
Solids have the fastest sound speed, liquids have the slowest sound speed, and gases have the slowest sound speed. This is because solids allow for faster energy transmission because their particles are closer together.
Depending on the characteristics of the new medium, a wave’s speed varies when it enters it. A sound wave, for example, moves more quickly through water than through air.