As a wave propagates:
(a) the wave intensity remains constant for a plane wave
(b) the wave intensit

Q: Why does the intensity of a plane wave remain constant? Doesn't it lose energy due to medium absorption?

In ideal wave propagation (which is assumed unless specified otherwise), we neglect damping or absorption by the medium. In such an ideal case, because the rays of a plane wave are parallel, the wavefront area does not spread out, keeping the intensity constant. In real-world media, absorption would cause exponential decay, but geometrically it remains constant.

Q: What is the difference between 'intensity' and 'total intensity over a surface'?

'Intensity' ($I$) is a local quantity representing power per unit area ($W/m^2$). 'Total intensity over a surface' refers to the total power ($P$) passing through that entire surface, obtained by integrating the intensity over the surface area ($P = \oint I \cdot dA$ in Watts).

Q: How does intensity vary for a line source (cylindrical wave)?

For an infinite line source, the wavefronts are concentric cylinders of area $A = 2\pi r L$. Therefore, the intensity varies as $I = \frac{P}{2\pi r L} \propto \frac{1}{r}$. It decreases as the inverse of the distance, not the inverse square.

Q: Does the inverse square law apply to all types of waves?

No, the inverse square law ($I \propto 1/r^2$) only applies to spherical waves propagating from a localized point source in three-dimensional space. It does not apply to plane waves or cylindrical waves.

Q: Why is option (d) correct if intensity itself is decreasing with distance?

Option (d) refers to the total power (integrated intensity) over the entire spherical surface. While the intensity at any single point decreases as $1/r^2$, the total surface area of the sphere increases as $r^2$. These two effects perfectly cancel each other out, keeping the total power constant due to conservation of energy.

Question

As a wave propagates
(a) the wave intensity remains constant for a plane wave
(b) the wave intensity decreases as the inverse of the distance from the source for a spherical wave
(c) the wave intensity decreases as the inverse square of the distance from the source for a spherical wave
(d) total intensity of the spherical wave over the spherical surface centred at the source remains constant at all times

Options:

(A) the wave intensity remains constant for a plane wave

(B) the wave intensity decreases as the inverse of the distance from the source for a spherical wave

(C) the wave intensity decreases as the inverse square of the distance from the source for a spherical wave

(D) total intensity of the spherical wave over the spherical surface centred at the source remains constant at all times

Sigma Percentile · Accepted Answer

### Introduction to Wave Propagation and Intensity

When a wave propagates through a medium, it acts as a carrier of **energy** and **momentum**.

To quantify how this energy is distributed in space, we define a fundamental physical quantity called **Wave Intensity ($I$)**.

Wave intensity is defined as the average energy transported by the wave per unit time (which is power, $P$) crossing a unit area ($A$) perpendicular to the direction of wave propagation:

$$I = \frac{P}{A}$$

Depending on the geometry of the source, waves can propagate in different shapes, leading to different types of wavefronts.

Let us analyze the three primary wave geometries: **plane waves**, **spherical waves**, and **cylindrical waves** to evaluate the given options.

---

### Analyzing Plane Waves

A plane wave is a wave in which the wavefronts are flat, parallel planes perpendicular to the direction of propagation.

Imagine a beam of light from a highly collimated laser. The rays of energy travel parallel to each other.

Because the rays are parallel, the cross-sectional area $A$ of the beam does not change as the wave travels:

$$A_1 = A_2 = 	ext{constant}$$

Since the power $P$ carried by the wave is constant and the area $A$ remains constant, the intensity $I$ must also remain constant:

$$I = \frac{P}{A} = 	ext{constant}$$

This confirms that **the wave intensity remains constant for a plane wave**. Therefore, **Option (a) is correct**.

---

### Analyzing Spherical Waves

Now, let us consider a point source $S$ that emits sound or light waves uniformly in all directions.

The wavefronts produced by a point source are concentric spheres expanding outwards.

At a distance $r$ from the source, the energy emitted by the source is distributed over the surface area of a sphere of radius $r$:

$$A = 4\pi r^2$$

Substituting this area into our intensity formula gives:

$$I = \frac{P}{4\pi r^2}$$

Since the power $P$ of the source is constant, we find that the intensity is inversely proportional to the square of the distance from the source:

$$I \propto \frac{1}{r^2}$$

This is known as the **Inverse Square Law**.

* This proves that the intensity decreases as the **inverse square** of the distance, making **Option (c) correct**.
* Consequently, **Option (b) is incorrect** because it states that intensity decreases as the simple inverse of the distance ($I \propto 1/r$).

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### Total Intensity over a Spherical Surface

Let us evaluate Option (d), which talks about the "total intensity over the spherical surface".

In physics, when we refer to the total energy crossing a closed surface per unit time, we are talking about the **total power ($P_{	ext{total}}$)**.

Mathematically, this is the surface integral of the intensity over the sphere:

$$P_{	ext{total}} = \oint I \cdot dA = I 	imes (4\pi r^2)$$

Substituting the expression for $I$ of a spherical wave:

$$P_{	ext{total}} = \left(\frac{P}{4\pi r^2}ight) 	imes (4\pi r^2) = P$$

Since the power $P$ of the source is constant, the total power crossing any concentric sphere is constant and independent of the radius $r$.

This is a direct consequence of the **Law of Conservation of Energy**—all the energy emitted by the source must pass through any closed sphere surrounding it, regardless of its size.

Therefore, **Option (d) is correct**.

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### Summary of Correct Options

By systematically applying the definition of wave intensity and geometric principles, we have determined that:
* **Option (a)** is correct because plane wave area is constant.
* **Option (c)** is correct because spherical wave area scales as $r^2$, leading to $I \propto 1/r^2$.
* **Option (d)** is correct because total energy is conserved across any closed wavefront.

Thus, the correct options are **(a), (c), and (d)**.

Select Answer:

Visualized Solution

Introduction to Wave Intensity

Analyzing Plane Waves

Intensity of a Plane Wave

Analyzing Spherical Waves

Deriving the Inverse Square Law

Total Intensity over Spherical Surface

Summary and Conclusion

The Sigma Insight: Wave Equation and Wave Speed

Introduction to Wave Propagation and Intensity

Analyzing Plane Waves

Analyzing Spherical Waves

Total Intensity over a Spherical Surface

Summary of Correct Options

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