Animated Solution for Physics - Waves: A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe,

Imagine you are standing at the entrance of a long, dark tunnel. You take a deep breath and shout into it. A fraction of a second later, your voice bounces back to you. This simple phenomenon of an echo is something we have all experienced, but the underlying physics of how sound waves reflect from boundaries is incredibly rich and elegant.

In this article, we will dive deep into the physics of acoustic boundaries, exploring how a high-pressure pulse behaves when it reaches the end of a pipe. This is not just a theoretical exercise; it is a classic concept tested in competitive exams like JEE Advanced.

Before we analyze the reflection, let's understand what a sound wave actually is. Sound is a longitudinal pressure wave. When you blow into a pipe, you create a region where air molecules are crowded together. This region of high density and high pressure is called a compression.

Conversely, a region where molecules are spread apart is called a rarefaction, which corresponds to low pressure. As the wave travels, these compressions and rarefactions propagate down the pipe.

When we talk about a "high-pressure pulse," we are talking about a single compression traveling through the medium. What happens when this compression reaches the end of the pipe? The answer depends entirely on whether the end is open or closed.

A Gateway to Freedom

Let's first consider the case where the other end of the pipe is open to the atmosphere.

The atmosphere is a massive reservoir of air at a constant pressure ($P_{\text{atm}}$). Because the open end is in direct contact with this vast reservoir, the pressure at the very boundary is forced to remain equal to $P_{\text{atm}}$. In other words, the pressure variation at the open end must always be zero. This makes the open end a pressure node.

When our high-pressure compression pulse reaches this open end, the compressed air molecules suddenly find themselves free to expand into the open atmosphere. Because there is no rigid wall to stop them, they rush outward.

Due to their inertia, the molecules overshoot their equilibrium positions. This outward rush leaves behind a region of extremely low pressure—a rarefaction—inside the pipe near the exit.

This low-pressure region then begins to travel back up the pipe. Physically, this means the incoming high-pressure pulse has reflected as a low-pressure pulse.

In wave mechanics, we describe this inversion as a phase change of $\pi$ radians (or $180^{\circ }$).

Thus, if the other end of the pipe is open, a low-pressure pulse starts traveling back up the pipe. This confirms that Option (b) is correct.

A Rigid Barrier

Now, let's look at the second scenario: the other end of the pipe is closed by a rigid wall.

At a closed end, the air molecules are physically blocked by the wall. They cannot move forward. This means the displacement of the air molecules at the boundary must be zero, making the closed end a displacement node.

Because the molecules cannot move, when the high-pressure compression pulse hits the wall, the molecules pile up against it. The pressure at the wall builds up to a maximum, making the closed end a pressure antinode.

Since the wall is rigid and cannot deform, it exerts an equal and opposite force back on the air molecules. The compressed air has nowhere to go but to bounce straight back into the pipe.

Because there is no expansion into an open space, the high-pressure state is preserved. The compression reflects directly as a compression.

In wave mechanics, reflection from a rigid boundary (a denser medium) occurs with no phase change ($\Delta \varphi =0$).

Thus, if the other end of the pipe is closed, a high-pressure pulse starts traveling back up the pipe. This confirms that Option (d) is correct.

Let's review the options provided in the question: - (a) a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is open. (Incorrect, it reflects as a low-pressure pulse) - (b) a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is open. (Correct) - (c) a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed. (Incorrect, it reflects as a high-pressure pulse) - (d) a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed. (Correct)

Therefore, the correct options are (b) and (d).

To master this concept for JEE, keep this simple summary table in mind:

Always remember to distinguish between pressure waves and displacement waves, as their phase changes are exactly opposite! With this conceptual clarity, you are ready to tackle any boundary value problem in acoustics.