Equation of a plane progressive wave is given by $y = 0.6\sin 2\pi\left(t - \dfrac{x}{2}\right)$. On reflection from a denser medium its amplitude becomes 2/3 of the amplitude of the incident wave. The equation of the reflected wave is (a) $y = 0.6\sin 2\pi\left(t + \dfrac{x}{2}\right)$ (b) $y = -0.4\sin 2\pi\left(t + \dfrac{x}{2}\right)$ (c) $y = 0.4\sin 2\pi\left(t + \dfrac{x}{2}\right)$ (d) $y = -0.4\sin 2\pi\left(t - \dfrac{x}{2}\right)$.

Equation of a plane progressive wave is given by y – Class 11 Physics

NCERT Exemplar · Q8

y = 0.6\sin 2\pi\left(t - \dfrac{x}{2}\right)

(a)

y = 0.6\sin 2\pi\left(t + \dfrac{x}{2}\right)

(b)

y = -0.4\sin 2\pi\left(t + \dfrac{x}{2}\right)

(c)

y = 0.4\sin 2\pi\left(t + \dfrac{x}{2}\right)

(d)

y = -0.4\sin 2\pi\left(t - \dfrac{x}{2}\right)

Sikkim CbseMCQ· 1mImportance★★★★★est

57% · 33/58 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Wave Speed on String

Wave Speed on a String – From Intuition to Formula

Imagine you and a friend hold a long, taut rope between you. If you give your end a quick flick upward, a bump travels along the rope toward your friend. That bump is a wave, and the speed at which it moves is the wave speed.

Now ask yourself: what determines how fast that bump travels? Two things stand out from everyday experience:

Tension – If you pull the rope tighter, the bump zips along faster. A loose rope makes the wave crawl.
Mass – If the rope is heavy (like a thick clothesline), the wave moves slower than on a light, thin string under the same tension.

So wave speed increases with tension and decreases with the "heaviness" of the string. That's the core intuition.

The Precise Statement

For a wave traveling along a stretched string, the wave speed $v$ is given by:

v = \sqrt{\frac{T}{\mu}}

where:

$T$ is the tension in the string (in newtons, N)
$\mu$ is the linear mass density – the mass per unit length of the string (in kg/m)

v = \sqrt{\frac{T}{\mu}}

This formula is exact for an ideal string (perfectly flexible, no stiffness, no damping). It comes from solving the wave equation for a string, but you can understand it physically.

Why the Square Root? A Quick Physical Argument

Think of a small segment of the string. The tension provides the restoring force that tries to straighten the string when it's bent. A higher tension means a stronger restoring force, so the wave accelerates faster – hence higher speed.

The mass per unit length $\mu$ is the inertia of the string. A heavier string resists acceleration more, so the wave slows down.

The square root appears because the relationship between force, mass, and acceleration isn't linear when you derive it properly. But the key takeaway is:

Important

Wave speed on a string depends only on the string's tension and its linear density – not on the frequency or amplitude of the wave.

This is a surprising and important result. Whether you send a slow, gentle ripple or a fast, sharp pulse, both travel at the same speed on the same string.

A Simple Example

A steel guitar string has $\mu = 0.002\ \text{kg/m}$ and is under tension $T = 100\ \text{N}$ . What is the wave speed?

v = \sqrt{\frac{100}{0.002}} = \sqrt{50000} \approx 224\ \text{m/s}

That's about half the speed of sound in air – fast enough that the wave reaches the other end almost instantly.

Common Mistakes to Avoid

Watch out

Do not confuse wave speed with the speed of the string's particles. The string itself moves up and down (transverse motion), but the wave travels horizontally. These are different speeds.
Wave speed does NOT depend on frequency. Changing how fast you flick your hand changes the frequency, but the wave still travels at $v = \sqrt{T/\mu}$ .
Tension is not the same as force applied at the end. If the string is under tension $T$ everywhere (ideal case), that's the value you use – not the force you apply to create the wave.

Where This Formula Comes From (A Glimpse) …

Start your 14-day free trial to unlock the full solution →

When a wave reflects from a denser medium, it undergoes a phase change of $\pi$ (inverts) and its amplitude becomes $\frac{2}{3} \times 0.6 = 0.4$ ; the direction reverses (changing the sign of $x$ ), giving $y = -0.4\sin 2\pi\left(t + \frac{x}{2}\right)$ .

Understanding Wave Reflection from a Denser Medium

When a wave traveling along a string (or any medium) encounters a boundary with a denser medium, two key changes occur:

Direction reversal: The reflected wave travels back in the opposite direction. If the incident wave moves in the $+x$ direction, the reflected wave moves in the $-x$ direction.

Phase inversion: Reflection from a denser medium (a fixed or rigid boundary) introduces a phase change of $\pi$ radians. Physically, this means the wave "flips" — a crest reflects as a trough. Mathematically, this multiplies the wave function by $-1$ .

The incident wave equation $y = 0.6\sin 2\pi\left(t - \frac{x}{2}\right)$ describes a wave traveling in the $+x$ direction with amplitude $A_i = 0.6$ .

Step-by-Step Construction of the Reflected Wave

Find the new amplitude The problem states the reflected amplitude is $\frac{2}{3}$ of the incident amplitude:

A_r = \frac{2}{3} \times 0.6 = 0.4

Reverse the direction of propagation

The incident wave has the form $\sin 2\pi\left(t - \frac{x}{2}\right)$ , where the minus sign indicates motion in the $+x$ direction (as $t$ increases, constant phase requires $x$ to increase).

For a wave traveling in the $-x$ direction, we replace $x$ with $-x$ in the argument:

\sin 2\pi\left(t - \frac{-x}{2}\right) = \sin 2\pi\left(t + \frac{x}{2}\right)

Apply the phase inversion …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ7 — A sound wave is passing through air column in the form of compression …→Next questionQ9 — A string of mass 2.5 kg is under a tension of 200 N. The length of the…

Q.Equation of a plane progressive wave is given by y=0.6sin⁡2π(t−x2)y = 0.6\sin 2\pi\left(t - \dfrac{x}{2}\right)y=0.6sin2π(t−2x​). On reflection from a denser medium its amplitude becomes 2/3 of the amplitude of the incident wave. The equation of the reflected wave is

Understanding Wave Reflection from a Denser Medium

Step-by-Step Construction of the Reflected Wave

🎓Unlock everything free for 14 days

Q.Equation of a plane progressive wave is given by $y = 0.6\sin 2\pi\left(t - \dfrac{x}{2}\right)$ . On reflection from a denser medium its amplitude becomes 2/3 of the amplitude of the incident wave. The equation of the reflected wave is

Unlock everything free for 14 days