The General Equation of a Progressive Wave

1. From an Oscillating Particle to a Traveling Wave

Imagine you have a long rope. If you start shaking one end ($x=0$) up and down in simple harmonic motion, you'll see a wave travel down the rope. This is a progressive wave—a disturbance that travels or "progresses" through a medium.

The core idea is this: every particle on the rope performs the same SHM as the first particle, but with a time delay. A particle at some position $x$ will do exactly what the particle at the origin did, just a little bit later.

Let's build the equation step-by-step:

1. The Source: The particle at the origin ($x=0$) oscillates according to the SHM equation. Its displacement at any time $t$ is:
   $y(0,t)=A\sin (\omega t)$

2. The Propagation: The wave travels along the positive x-axis with a constant speed, $v$.

3. The Time Lag: How long does it take for the disturbance to travel from the origin to a point $P$ at a distance $x$?
   $\text{Time delay}=\frac{\text{Distance}}{\text{Speed}}=\frac{x}{v}$

4. The Motion at Point P: The particle at point $P$ will start its oscillation $\frac{x}{v}$ seconds after the particle at the origin. Therefore, its displacement at time $t$ must be the same as the displacement of the origin particle at the earlier time $t-\frac{x}{v}$.

5. The Wave Equation: To find the displacement at $x$ at time $t$, which we call $y(x,t)$, we take the equation for the source and substitute $t$ with $(t-\frac{x}{v})$:
   $y(x,t)=y\left(0,t-\frac{x}{v}\right)$
   $y(x,t)=A\sin \left[\omega \left(t-\frac{x}{v}\right)\right]$

This is the fundamental equation for a one-dimensional progressive wave traveling in the positive x-direction.

To see this derivation explained with clear visuals, watch the following video. It builds the logic exactly as we just did.

2. The Many Faces of the Wave Equation

The equation we derived, $y(x,t)=A\sin [\omega (t-x/v)]$, is correct but can be written in several other forms that are more common in physics problems. Let's explore them.

Introducing the Wave Number, $k$

Let's expand the argument of the sine function:
$y(x,t)=A\sin \left(\omega t-\frac{\omega }{v}x\right)$
Now, let's examine the term $\frac{\omega }{v}$. We know the angular frequency $\omega =\frac{2\pi }{T}$ (where $T$ is the time period) and the wave speed $v=\frac{\lambda }{T}$ (where $\lambda$ is the wavelength).

Substituting these gives:
$\frac{\omega }{v}=\frac{2\pi /T}{\lambda /T}=\frac{2\pi }{\lambda }$
This important quantity, $\frac{2\pi }{\lambda }$, is called the angular wave number or simply the wave number, denoted by $k$.
$k=\frac{2\pi }{\lambda }$
The wave number $k$ has units of rad/m. It represents the spatial frequency of the wave—it tells you how many radians of phase change occur per unit distance.

By substituting $k$ back into our equation, we get the most common form of the wave equation:
$y(x,t)=A\sin (\omega t-kx)$

Here is a visual summary of this derivation logic.

Other Common Forms

All of the following equations describe the same wave. Being able to recognize and convert between them is a key skill for NEET.

Starting from $y(x,t)=A\sin (\omega t-kx)$:

1. Substitute $\omega =\frac{2\pi }{T}$ and $k=\frac{2\pi }{\lambda }$:
   $y(x,t)=A\sin \left(\frac{2\pi }{T}t-\frac{2\pi }{\lambda }x\right)=A\sin \left[2\pi \left(\frac{t}{T}-\frac{x}{\lambda }\right)\right]$

2. Factor out $k=\frac{2\pi }{\lambda }$ and use $v=\frac{\omega }{k}$:
   $y(x,t)=A\sin \left[k\left(\frac{\omega }{k}t-x\right)\right]=A\sin [k(vt-x)]$

Direction of Propagation

The sign between the time ($t$) and position ($x$) terms determines the wave's direction.

• Negative sign (ωt - kx or kx - ωt): The wave travels in the positive x-direction. For a point of constant phase (e.g., a crest), as time t increases, position x must also increase to keep the phase (ωt - kx) constant.
• Positive sign (ωt + kx): The wave travels in the negative x-direction. As t increases, x must decrease to keep the phase (ωt + kx) constant.

Finally, we can add an initial phase constant, $\varphi$, to account for the starting condition of the wave at $x=0$ and $t=0$. The most general form is:
$y(x,t)=A\sin (kx\pm \omega t+\varphi )$

The following resource provides an excellent formal summary of these mathematical relationships.

3. An Alternative Perspective

There's another intuitive way to build the wave equation, which complements the time-delay method. You can start by taking a "snapshot" of the wave in space at $t=0$ and then make that snapshot move. The Khan Academy video below explains this approach brilliantly.

Both derivation methods lead to the same result and reinforce your understanding of the relationship between space, time, and phase in a wave. For quick revision of all the forms, the following notes are very helpful.

Conclusion

Today we have built the mathematical framework for describing waves. By starting with the simple harmonic motion of a single particle and considering the time it takes for that motion to propagate, we derived the general equation for a one-dimensional progressive wave.

Key Takeaways:

• A wave is a propagating oscillation. The motion of any particle at position $x$ is a time-delayed version of the motion at the origin.
• The general equation for a sinusoidal wave can be expressed in several equivalent forms. The most common is:
  $y(x,t)=A\sin (kx\pm \omega t+\varphi )$
• Key parameters are Amplitude ($A$), Wave Number ($k=2\pi /\lambda$), and Angular Frequency ($\omega =2\pi /T$).
• The sign between the $kx$ and $\omega t$ terms determines the direction of travel: '+' for the negative direction and '-' for the positive direction.

In our next lesson, we will put this equation to practical use. We will learn how to construct the specific equation of a wave from its physical parameters (amplitude, wavelength, frequency) and how to extract these parameters from a given equation—a crucial skill for solving NEET-style problems.