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Wize University Physics Textbook (Master) > Waves: Mechanical

Traveling Waves

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General Definition of a Traveling Wave



From the mathematical point of view, any function that can move is a wave! Physically, a wave is the propagation of energy in space. The shape of this wave in space is shown by a function, f(x,t)f(x,t), which is called the wave function and it depends on time and position in space.

Example:












In physics, the wave function should satisfy the following equation which is known as the wave equation:


∂2f∂x2−1v2∂2f∂t2=0\dfrac{\partial^2f}{\partial x^2}-\dfrac{1}{v^2}\dfrac{\partial^2f}{\partial t^2}=0


where in the above equation, vvis the speed at which the wave is traveling in space and it is called wave speed.


Wize Concept
It can be shown that any function in the form of f(x±vt)f(x\pm vt) is a solution to the above equation, and as a result, a wave function.


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Example: A traveling wave moving toward right
















Example: A traveling wave moving toward left















Sinusoidal Waves


One of the important family of solutions to the wave equation is the oscillating wave functions that could be described by sine or cosine functions. These wave functions are known as sinusoidal waves and they have the following mathematical description:



 f(x,t)=Asin⁡[k(x±vt)+ϕ] \boxed{ \ f(x,t)=A \sin[k(x\pm vt)+\phi] \ }

where:
  • AA is the amplitude of this oscillating function.
  • kk is known as the wave number and is the number of full oscillations in space per unit of time.
  • vv is the speed at which the wave is traveling in space.
  • ϕ\phi is known as the phase constant and tells us the initial condition at which the wave started.

The speed of any sinusoidal wave is given by:
 v=λ f \boxed{\ v=\lambda\ f \ }












Wize Tip
Note that we can simply write down above function in terms of cosine function by adjusting the phase constant using trigonometric relationships.

There are two important property of these types of the waves:

  1. These functions are periodic functions with a period of 2π2\pi. So, the pattern repeats itself forever.
  2. In this sinusoidal functions, all particles in the medium undergo a simple harmonic motion. The collective motion of all these particles creates a wave which is traveling in space.
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Transverse vs Longitudinal Waves:

There are two types of traveling wave:

  1. A wave is called transverse wave if the direction of motion of particles in the medium is perpendicular to direction at which the wave is propagating in space. Example: Wave in a rope.
  2. A wave is called longitudinal wave if the direction of motion of particles in the medium is along the direction at which the wave is propagating in space. Example: pinching a slinky! or sound waves












Example: Properties of Waves

A traveling wave completes 2525 vibrations in 20.020.0 seconds. You also observe that the peak (crest) of the wave travels a distance of 2.002.00 meters in 1.501.50 seconds. Find the frequency, angular frequency, period, speed, wavelength and wave number.

Frequency, angular frequency, period:

We have 2525 vibrations (or oscillations) in 2020s, we need frequency which is the number of oscillations per unit time, therefore:

f=2520=1.25f=\dfrac{25}{20}=1.25 (Hz or s-1)

The angular frequency is the angle covered per unit time, and since one oscillation corresponds to an angle of 2π2\pi, we have:

ω=2πf=2π(1.25)=2.5π\omega=2\pi f = 2\pi (1.25)=2.5 \pi (rad/s)

Period and frequency are reciprocals of each other, so the period is:

T=1f=2025=0.8T=\dfrac{1}{f}=\dfrac{20}{25}=0.8 (s)

Speed:

Let's use the basic definition of speed = distance/time to get:

v=21.5=1.33v=\frac{2}{1.5}=1.33 (m/s)

Wavelength and wave number:

Using the definition of speed we have:

v=λf   →   λ=vf=1.331.25=1.07v=\lambda f\ \ \ \to\ \ \ \lambda=\frac{v}{f}=\frac{1.33}{1.25}=1.07 (m)

The wave number is the number of complete oscillations (in radians) per unit distance, therefore:

k=2πλ=2π1.07=1.875 πk=\dfrac{2\pi}{\lambda}=\dfrac{2\pi}{1.07}=1.875 \ \pi (rad/m)

Practice: Properties of Waves


The wavelength of a transverse wave is 1.501.50 m. It is observed that 1010 crests of the wave pass a specified point every 12.012.0 seconds. What is the speed of the wave? How does the wave speed change if the frequency is doubled in the same medium? How long does it take the wave to travel 15.015.0 m?