A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this w

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2.06 \times 10^{4}

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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) …

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The speed of a transverse wave on a string depends only on tension and linear mass density. For this steel wire, the wave speed is $v = \sqrt{T/\mu} \approx 343 \, \text{m/s}$ .

The key idea is that a transverse wave travels along a stretched string because the tension provides the restoring force. The wave speed is determined by two properties of the string: how stiffly it resists being stretched (tension $T$ ) and how much inertia each segment has (mass per unit length $\mu$ ). A heavier string moves more sluggishly; a tighter string snaps back faster. The formula $v = \sqrt{T/\mu}$ captures this balance exactly.

Let’s work through the numbers.

Find the linear mass density $\mu$ . The wire’s total mass is $m = 2.10 \, \text{kg}$ and its length is $L = 12 \, \text{m}$ . The mass per unit length is simply the mass divided by the length:

\mu = \frac{m}{L} = \frac{2.10}{12} = 0.175 \, \text{kg/m}.

Apply the wave speed formula. The tension is $T = 2.06 \times 10^{4} \, \text{N}$ . The speed of a transverse wave on a string under tension is:

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

Substituting the values:

v = \sqrt{\frac{2.06 \times 10^{4}}{0.175}}.

Simplify the calculation. First compute the ratio inside the square root:

\frac{2.06 \times 10^{4}}{0.175} = \frac{20600}{0.175}.

Dividing: $0.175 \times 117714.2857 \approx 20600$ , so:

\frac{20600}{0.175} = 117714.2857 \ldots

More neatly, multiply numerator and denominator by 1000 to avoid decimals:

$\frac{20600}{0.175} = \frac{20600 \times 1000}{175} = \frac{20600000}{175}.$ …

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Q.A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this wire when a tension of 2.06×1042.06 \times 10^{4}2.06×104 N is applied?

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Q.A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this wire when a tension of $2.06 \times 10^{4}$ N is applied?

Unlock everything free for 14 days