Speed of Wave on String — Revision Notes

⚡ 30-Second Revision

Wave Speed Formula: — $v = \sqrt{T/\mu}$
Tension (T): — Force stretching the string, in Newtons (N).
Linear Mass Density ($\mu$): — Mass per unit length, $\mu = m/L$, in kg/m.
Units: — Ensure $m$ in kg, $L$ in m, $T$ in N, $\mu$ in kg/m, $v$ in m/s.
Dependence: — $v \propto \sqrt{T}$, $v \propto 1/\sqrt{\mu}$.
Independence: — $v$ is independent of amplitude and frequency.
General Wave Equation: — $v = f\lambda$.
Hanging String: — $T = \mu x g$ at distance $x$ from bottom, so $v = \sqrt{gx}$.

2-Minute Revision

The speed of a transverse wave on a stretched string is a crucial concept for NEET. It's determined by the string's physical properties, specifically its tension ($T$) and linear mass density ($\mu$).

The fundamental formula is $v = \sqrt{T/\mu}$. Remember that tension is the restoring force, so higher tension means faster waves ($v \propto \sqrt{T}$). Linear mass density (mass per unit length, $\mu = m/L$) represents inertia; a heavier string (higher $\mu$) means slower waves ($v \propto 1/\sqrt{\mu}$).

Crucially, the wave speed is independent of the wave's amplitude or frequency. Always convert units to SI (mass in kg, length in m) before calculation. For problems involving a vertically hanging string, remember that tension varies along its length, increasing from bottom to top.

At a distance $x$ from the bottom, $T = \mu x g$, leading to $v = \sqrt{gx}$. Don't forget the general wave relation $v = f\lambda$ which often combines with this topic.

5-Minute Revision

Mastering the speed of a transverse wave on a string is essential for NEET. The core principle is that wave speed ($v$) depends on the medium's elastic property (tension, $T$) and its inertial property (linear mass density, $\mu$). The formula is $v = \sqrt{T/\mu}$.

Tension (T): This is the force stretching the string, measured in Newtons. Higher tension means a stronger restoring force, causing the wave to travel faster. For example, if tension is quadrupled, speed doubles ($v \propto \sqrt{T}$). If a mass $M$ hangs from a string, $T=Mg$.

**Linear Mass Density ($\mu$):** This is the mass per unit length of the string, $\mu = m/L$, measured in kg/m. It represents the string's inertia. A thicker or denser string has higher $\mu$, leading to slower wave speeds. If $\mu$ is quadrupled, speed halves ($v \propto 1/\sqrt{\mu}$). If string material has volume density $\rho$ and radius $r$, then $\mu = \rho \pi r^2$.

Key Points:

Units: — Always use SI units: $T$ in N, $m$ in kg, $L$ in m, $\mu$ in kg/m, $v$ in m/s.
Independence: — Wave speed on a string is independent of its amplitude and frequency. These are determined by the source, not the medium.
General Wave Equation: — Remember $v = f\lambda$. This allows you to find frequency or wavelength if speed is known.
Vertically Hanging String: — Tension is not uniform. At a distance $x$ from the bottom, $T = \mu x g$. Thus, $v = \sqrt{xg}$. This means the wave accelerates as it travels up.

Example: A $1\,\text{m}$ long string has a mass of $10\,\text{g}$ and is under $100\,\text{N}$ tension. Find wave speed.

1
$\mu = (10 \times 10^{-3}\,\text{kg}) / 1\,\text{m} = 0.01\,\text{kg/m}$.
2
$v = \sqrt{100\,\text{N} / 0.01\,\text{kg/m}} = \sqrt{10000}\,\text{m/s} = 100\,\text{m/s}$.

Practice ratio problems and conceptual questions about factors affecting speed to solidify your understanding.

Prelims Revision Notes

Speed of Wave on String: NEET Revision Notes

1. Fundamental Formula:

* The speed ($v$) of a transverse wave on a stretched string is given by: $v = \sqrt{T/\mu}$ * Where: * $T$ = Tension in the string (in Newtons, N) * $\mu$ = Linear mass density of the string (in kilograms per meter, kg/m)

2. Key Parameters and Their Influence:

* Tension (T): * Represents the elastic restoring force in the string. * Higher tension ($T$) leads to higher wave speed ($v \propto \sqrt{T}$). If $T$ is quadrupled, $v$ doubles. * Often, tension is provided indirectly, e.

g., by a hanging mass $M$, so $T = Mg$. * **Linear Mass Density ($\mu$):** * Defined as mass per unit length: $\mu = m/L$ (where $m$ is total mass, $L$ is total length). * Represents the inertia of the string segments.

* Higher linear mass density ($\mu$) leads to lower wave speed ($v \propto 1/\sqrt{\mu}$). If $\mu$ is quadrupled, $v$ halves. * For a string of uniform material (volume density $\rho$) and circular cross-section (radius $r$), $\mu = \rho \pi r^2$.

3. Independence from Amplitude and Frequency:

* Crucially, the speed of a wave on an ideal string is determined *only* by the properties of the medium ($T$ and $\mu$). * It does not depend on the amplitude or frequency of the wave. These are characteristics of the wave source.

4. Relationship with Frequency and Wavelength:

* The general wave equation $v = f\lambda$ also applies. * Combining: $f\lambda = \sqrt{T/\mu}$. This is useful for problems involving pitch (frequency) of musical instruments.

5. Vertically Hanging String:

* For a string suspended vertically, tension is not uniform. * Tension at a distance $x$ from the bottom end is due to the weight of the string segment below it: $T(x) = \mu x g$. * Therefore, the wave speed at a distance $x$ from the bottom is $v(x) = \sqrt{T(x)/\mu} = \sqrt{(\mu x g)/\mu} = \sqrt{xg}$. * The wave accelerates as it travels upwards.

6. Unit Conversions:

* Always use SI units: Mass in kg, Length in m, Tension in N, Linear mass density in kg/m, Speed in m/s. * Example: $1\,\text{g} = 10^{-3}\,\text{kg}$, $1\,\text{cm} = 10^{-2}\,\text{m}$.

7. Common Traps:

* Forgetting to convert units. * Confusing total mass ($m$) with linear mass density ($\mu$). * Assuming speed depends on amplitude or frequency. * Incorrectly applying square roots in ratio problems.

Vyyuha Quick Recall

Tension Makes Us Very Speedy! (Tension, Mass per unit length, Velocity, Square root)