14.3 Displacement relation in a progressive wave

Progressive wave

A wave that travels from one point of the medium to another is called a progressive wave. It can be transverse or longitudinal. A wave which travels continuously in a medium in the same direction without a change in its amplitude is called a travelling wave or a progressive wave.

A progressive (or travelling) wave is a continuous disturbance that carries energy through a medium in one direction without changing its amplitude, defined by \(y(x, t )=a \sin (k x-\omega t+\phi)\). It can be transverse (displacement perpendicular to direction) or longitudinal (parallel).

Key Aspects of Progressive Waves

Transverse Wave Example: A sine wave on a string, where \(y(x, t)\) is the vertical displacement of a particle at position \({x}\) and time \({t}\).
Direction: The term \((k x-\omega t)\) indicates travel in the positive \(x\) -direction, while (\(k x +\omega t)\) indicates travel in the negative direction.

Direction of Wave:

To understand why the sign between \(k x\) and \(\omega t\) determines the direction of the wave, we have to look at a single point on the wave (like a crest) and see how it must move to stay a “crest” as time passes.

Think of the argument of the sine function, \((k x \pm \omega t)\), as the phase of the wave. For a specific feature of the wave (the peak, the trough, or any specific point) to persist, that phase value must remain constant.
Step 1: Traveling in the Positive Direction: (\(k x-\omega t\))
Imagine you are “riding” on the crest of a wave. For you to stay on that crest, the value of the argument must stay the same:
\(
k x-\omega t=\mathrm{constan} t
\)
As time \((t)\) increases, the term \(-\omega t\) becomes a larger negative number. To keep the total sum constant, the term \(k x\) must increase.
Since \(k\) is a positive constant, \(x\) must increase.
If \(x\) is increasing as \(t\) increases, the wave is moving to the right (positive \(x\)-direction).
Step 2: Traveling in the Negative Direction: \((k x+\omega t)\)
Now, look at the equation where the terms are added:
\(
k x+\omega t=\text { constant }
\)
Again, as time \((t)\) increases, the term \(+\omega t\) becomes a larger positive number. To keep the total sum from changing (to stay on that same crest), the term \(k x\) must decrease.
For \(k x\) to get smaller, \(x\) must move toward more negative values.
Therefore, the wave is moving to the left (negative \(x\)-direction).

Remark: To understand why the expression \(k x-\omega t\) must be constant, you have to stop thinking about the whole wave for a second and focus on just one specific feature-like the very tip of a single crest.
Identifying a “Feature”:
In the equation \(y(x, t)=a \sin (k x-\omega t)\), the displacement \(y\) depends entirely on the value inside the sine function. Let’s call that value the phase (\(\theta\)):
\(
\theta=k x-\omega t
\)
If you want to track a crest, you are looking for the point where the sine function is at its maximum. This happens whenever:
\(
\theta=\frac{\pi}{2}, \frac{5 \pi}{2}, \ldots
\)
To “stay” on that crest as time moves forward, that value (\(\pi / 2\)) cannot change. If the phase changed, you would slip off the crest and move down toward a trough. Therefore, for a “fixed phase point,” the argument must be constant.
Summary: We set it to constant because that is the mathematical definition of tracking a specific part of the wave. If \(k x-\omega t\) were changing, \(y\) would be changing, meaning you are looking at different parts of the wave (crests, troughs, and zeros) rather than following the motion of a single wave front.

The Mathematical “Snapshot” Analogy:

Think of it like a photograph \(f(x)\) of a wave.
To move a function to the right by a distance \(d\), you replace \(x\) with \((x-d)\). Since distance \(d=v t\), the moving wave becomes \(f(x-v t)\).
Substituting our wave constants \((v=\omega / k)\) :
\(
f(x-v t)=f\left(x-\frac{\omega}{k} t\right)
\)
Factor out the \(k\), and you get the familiar form: \(k(x)-\omega t\).

Mathematical Components:

a: Amplitude (maximum displacement).
\(k\) : Angular wavenumber \((2 \pi / \lambda)\).
\(\omega\) : Angular frequency \((2 \pi v)\).
\(\phi\) : Initial phase constant.
Particle Motion: Individual particles in the medium execute simple harmonic motion (SHM) as the wave passes.

Wave Parameters:

Wavelength (\(\lambda\)): Distance between two consecutive crests or troughs.
Frequency (\({f}\)): Number of complete waves passing a point per second.
Wave Speed \((v)\) : Defined as \(v=\frac{\omega}{k}=f \lambda\).

For convenience, we shall take the wave to be transverse so that if the position of the constituents of the medium is denoted by \(x\), the displacement from the equilibrium position may be denoted by \(y\). A sinusoidal travelling wave is then described by:
\(
y(x, t)=a \sin (k x-\omega t+\phi) \dots(i)
\)
The term \(\phi\) in the argument of sine function means equivalently that we are considering a linear combination of sine and cosine functions:
\(
y(x, t)=A \sin (k x-\omega t)+B \cos (k x-\omega t) \dots(ii)
\)
From Equations (i) and (ii),
\(
a=\sqrt{A^2+B^2} \text { and } \phi=\tan ^{-1}\left(\frac{B}{A}\right)
\)
To understand why Equation (i) represents a sinusoidal travelling wave, take a fixed instant, say \(t=t_0\). Then, the argument of the sine function in Equation (i) is simply \(k x+\) constant. Thus, the shape of the wave (at any fixed instant) as a function of \(x\) is a sine wave. Similarly, take a fixed location, say \(x=X_0\). Then, the argument of the sine function in Equation (i) is constant \(-\omega t\). The displacement \(y\), at a fixed location, thus, varies sinusoidally with time. That is, the constituents of the medium at different positions execute simple harmonic motion. Finally, as \(t\) increases, \(x\) must increase in the positive direction to keep \(k x-\omega t+\phi\) constant. Thus, Eq. (i) represents a sinusiodal (harmonic) wave travelling along the positive direction of the \(x\)-axis. On the other hand, a function
\(
y(x, t)=a \sin (k x+\omega t+\phi)
\)
represents a wave travelling in the negative direction of \(x\)-axis. Fig 15.6. below gives the names of the various physical quantities appearing in Eq. (i) that we now interpret.

Progressive wave in different Forms

A progressive wave (or traveling wave) is a disturbance that moves through a medium, continuously carrying energy and momentum away from the source. To describe this mathematically, we need an equation that tells us the displacement of a particle (\(y\)) based on both its position (\(x\)) and the time (\(t\)).
The Standard Wave Equation
The most common form of a simple harmonic progressive wave traveling in the positive \(x\) direction is:
\(
y(x, t)=A \sin (k x-\omega t+\phi)
\)
Where:
\(y\) : Displacement of the particle at position \(x\) and time \(t\).
\(A\): Amplitude (maximum displacement from equilibrium).
\(k\) : Angular Wave Number, defined as \(k=\frac{2 \pi}{\lambda}\) (measured in \(\mathrm{rad} / \mathrm{m}\)).
\(\omega\) : Angular Frequency, defined as \(\omega=2 \pi f=\frac{2 \pi}{T}\) (measured in rad/s).
\(\phi\) : Phase Constant (determines the initial displacement at \(x=0, t=0\)).

Derive Different Forms

Deriving the Period (\(T\)) Form: (\(\text { we assume the phase constant } \phi=0 \text {. }\))

\(
y=A \sin \left[\left(\frac{2 \pi}{\lambda}\right) x-\left(\frac{2 \pi}{T}\right) t\right]
\)
Factor out \(2 \pi\) :
\(
y=A \sin \left[2 \pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)\right]
\)
If we reverse the terms to represent the time-evolution (\(v t-x\) style), we get:
\(
y=A \sin \left[2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)\right]
\)

Deriving the Frequency (\(f\)) Form:

Since we know that frequency \(f=\frac{1}{T}\), we can simply replace \(\frac{1}{T}\) in the previous equation with \(f\)
\(
y=A \sin 2 \pi\left(f t-\frac{x}{\lambda}\right)
\)

Deriving the Velocity (\(v\)) Form:

Starting from the Period form again:
\(
y=A \sin \left[2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)\right]
\)
We want to introduce \(v\). We know that \(v=\frac{\lambda}{T}\), which means \(\frac{1}{T}=\frac{v}{\lambda}\). Substitute this into the equation:
\(
y=A \sin \left[2 \pi\left(\frac{v t}{\lambda}-\frac{x}{\lambda}\right)\right]
\)
Now, factor out the common denominator \(\lambda\) from both terms:
\(
y=A \sin \left[\frac{2 \pi}{\lambda}(v t-x)\right]
\)

Displacement Relation in a Progressive Wave

During the propagation of a wave through a medium, if the particles of the medium vibrate simple harmonically about their mean positions, then the wave is said to be plane progressive harmonic wave.

In a progressive wave, the displacement relation describes how the position of a particle changes over time as the wave passes through. It shows the relationship between the displacement of the particle from its equilibrium position and both the position along the wave (represented by \(x\) ) and the time (represented by \(t\) ).
Mathematically, the displacement relation for a sinusoidal (harmonic)  wave travelling in \(+x\)-direction is given by
\(
y(x, t)=A \sin (\omega t-k x+\phi) \dots(1)
\)
On the other hand, a wave travelling in \(-x\)-direction is given by
\(
y(x, t)=A \sin (\omega t+k x+\phi) \dots(2)
\)
Where, \(y(x, t)\) is the displacement of the particle at position \(x\) and time \(t\),
\(A\) is the amplitude, representing the maximum displacement from equilibrium,
\(k\) is the angular wave number, determining the spatial frequency of the wave, \(k=2 \pi / \lambda\)
\(\omega\) is the angular frequency, indicating how quickly the wave oscillates in time,
\(\phi\) is the phase constant, determining the initial position of the wave.
This relation shows how the displacement of a particle varies with both position and time along the wave. As the wave propagates, particles oscillate back and forth around their equilibrium positions according to this relation. It helps understand the behavior of waves and their interaction with particles in a medium.

Amplitude and Phase

In Eq. (1), since the sine function varies between 1 and -1, the displacement \(y(x, t)\) varies between \(A\) and \(-A\). We can take \(A\) to be a positive constant, without any loss of generality. Then, \(A\) represents the maximum displacement of the constituents of the medium from their equilibrium position. Note that the displacement \(y\) may be positive or negative, but \(A\) is positive. It is called the amplitude of the wave.

The quantity ( \(\omega t-k x+\phi\) ) appearing as the argument of the sine function in Eq. (1) is called the phase of the wave. Given the amplitude \(A\), the phase determines the displacement of the wave at any position and at any instant. Clearly \(\phi\) is the phase at \(x=0\) and \(t=0\). Hence, \(\phi\) is called the initial phase angle. By suitable choice of origin on the \(x\)-axis and the initial time, it is possible to have \(\phi=0\). Thus there is no loss of generality in dropping \(\phi\), i.e., in taking Eq. (1) with \(\phi=0\).

Wavelength and Angular Wave Number

The minimum distance between two points having the same phase is called the wavelength of the wave, usually denoted by \(\lambda\). For simplicity, we can choose points of the same phase to be crests or troughs. The wavelength is then the distance between two consecutive crests or troughs in a wave. Taking \(\phi=0\) in Eq. (1), the displacement at \(t=0\) is given by
\(
y(x, 0)=A \sin k x \dots(3)
\)
Since the sine function repeats its value after every \(2 \pi\) change in angle,
\(
\sin k x=\sin (k x+2 n \pi)=\sin k\left(x+\frac{2 n \pi}{k}\right)
\)
That is the displacements at points \(x\) and at \(x+\frac{2 n \pi}{k}\) are the same, where \(n=1,2,3, \ldots\) The least distance between points with the same displacement (at any given instant of time) is obtained by taking \(n=1. \lambda\) is then given by
\(
\lambda=\frac{2 \pi}{k} \quad \text { or } \quad k=\frac{2 \pi}{\lambda} \dots(4)
\)
\(k\) is the angular wave number or propagation constant; its SI unit is radian per metre or \(\mathrm{rad}  \mathrm{~m}^{-1 }\)

Period, Angular Frequency and Frequency

\(\omega\) is called the angular frequency. Here,
\(
\omega=2 \pi f=\frac{2 \pi}{T} \quad \text { and } \quad f=\frac{1}{T}
\)
where, \(f\) is the normal frequency (natural frequency) of oscillations and \(T\) is the time period of oscillations. SI unit of \(\omega\) is radian/sec while that of \(f\) is Hz or \(\mathrm{s}^{-1}\).

Wave Speed (\(v\))

Wave speed \(v=\frac{\text { coefficient of } t}{\text { coefficient of } x}=\frac{\omega}{k}\)
But \(\omega=2 \pi f\) and \(k=\frac{2 \pi}{\lambda}\)
\(v=f \lambda=\frac{\lambda}{T}\)

Notes: (i) Equation of the form \(y(x, t)=f\) (at \(\pm b x\) ) represents progressive or travelling waves. The plus sign denotes a wave traveling in the negative direction of the \(x\) axis, and the minus sign a wave traveling in the positive direction.
Some common progressive wave equations are
\(y=A \log (a t+b x)\)
\(y=\sqrt{(a x+b t)}\)
\(y=(a x-b t)^2\)
\(y=A \sin (a x-b t)^2\)
\(y=a \cos ^2(\omega t-k x)\)
\(y=a \cos \Delta \omega t \sin (\omega t-k x)\)

Example 1: A wave travelling along a string is described by,
\(
y(x, t)=0.005 \sin (80.0 x-3.0 t),
\)
in which the numerical constants are in SI units \(\left(0.005 \mathrm{~m}, 80.0 \mathrm{rad} \mathrm{m}^{-1}\right.\), and \(3.0 \mathrm{rad} \mathrm{s}^{-1}\)). Calculate (a) the amplitude, (b) the wavelength, and (c) the period and frequency of the wave. Also, calculate the displacement \(y\) of the wave at a distance \(x=30.0 \mathrm{~cm}\) and time \(t=20 \mathrm{~s}\) ?

Solution: A sinusoidal travelling wave is then described by:
\(
y(x, t)=a \sin (k x-\omega t+\phi)
\)
On comparing this displacement equation with the above equation, (\(\phi=0\))
\(
y(x, t)=a \sin (k x-\omega t),
\)
we find
(a) the amplitude of the wave is \(0.005 \mathrm{~m}=5 \mathrm{~mm}\).
(b) the angular wave number \(k\) and angular frequency \(\omega\) are
\(
k=80.0 \mathrm{~m}^{-1} \text { and } \omega=3.0 \mathrm{~s}^{-1}
\)
We, then, relate the wavelength \(\lambda\) to \(k\):
\(
\begin{aligned}
\lambda & =2 \pi / k \\
& =\frac{2 \pi}{80.0 \mathrm{~m}^{-1}} \\
& =7.85 \mathrm{~cm}
\end{aligned}
\)
(c) Now, we relate \(T\) to \(\omega\) by the relation
\(
\begin{aligned}
T & =2 \pi / \omega \\
& =\frac{2 \pi}{3.0 \mathrm{~s}^{-1}} \\
& =2.09 \mathrm{~s}
\end{aligned}
\)
and frequency, \(v=1 / T=0.48 \mathrm{~Hz}\)
The displacement \(y\) at \(x=30.0 \mathrm{~cm}\) and time \(t=20 \mathrm{~s}\) is given by
\(
\begin{aligned}
y & =(0.005 \mathrm{~m}) \sin (80.0 \times 0.3-3.0 \times 20) \\
& =(0.005 \mathrm{~m}) \sin (-36+12 \pi) \\
& =(0.005 \mathrm{~m}) \sin (1.699) \\
& =(0.005 \mathrm{~m}) \sin \left(97^{\circ}\right) \simeq 5 \mathrm{~mm}
\end{aligned}
\)

Relationship between phase difference \((\phi)\) and path difference \((\Delta x)\):

The Wave Equation:
A simple harmonic wave traveling in the positive \(x\)-direction is represented by:
\(
y=A \sin \left(k x-\omega t+\theta_0\right)
\)
The phase of the wave at any point is the argument of the sine function:
\(
\Phi=k x-\omega t+\theta_0
\)
Defining Phase Difference:
Consider two points, \(x_1\) and \(x_2\), along the path of the wave at the same instant in time \((t)\). The phase at each point is:
\(\Phi_1=k x_1-\omega t+\theta_0\)
\(\Phi_2=k x_2-\omega t+\theta_0\)
The phase difference \((\phi)\) between these two points is:
\(
\begin{gathered}
\phi=\Phi_2-\Phi_1=\left(k x_2-\omega t+\theta_0\right)-\left(k x_1-\omega t+\theta_0\right) \\
\phi=k\left(x_2-x_1\right)
\end{gathered}
\)
Since the path difference is \(\Delta x=x_2-x_1\), we have:
\(
\phi=k \Delta x
\)
Introducing Wavelength (\(\lambda\)):
The propagation constant (or wave number) \(k\) is related to the wavelength \(\lambda\) by the formula:
\(
k=\frac{2 \pi}{\lambda}
\)
Substituting this value of \(k\) into our equation:
\(
\phi=\left(\frac{2 \pi}{\lambda}\right) \Delta x
\)
Final Relationships:
From this derivation, we can express the relationship in two ways depending on what you are solving for:
To find Phase Difference:
\(
\phi=\frac{2 \pi}{\lambda} \cdot \Delta x
\)
To find Path Difference:
\(
\Delta x=\frac{\lambda}{2 \pi} \cdot \phi
\)