How does increasing the tension of a string affect the speed of a wave traveling along it?

Increasing the tension increases the wave speed. This occurs because tension acts as the restoring force; a higher force pulls the string back to equilibrium faster, allowing the disturbance to propagate more quickly.

What is the mathematical relationship between wave speed ($v$), tension ($\tau$), and linear mass density ($\mu$)?

The relationship is given by the formula $v = \sqrt{\tau / \mu}$. This shows that speed is directly proportional to the square root of the tension and inversely proportional to the square root of the linear mass density.

If you double the frequency of a wave on a fixed string, what happens to the wave speed?

The wave speed remains unchanged. Wave speed is determined by the medium's properties (tension and density); doubling the frequency will simply result in the wavelength being halved to maintain the relationship $v = f\lambda$.

What is the difference between wave speed and particle speed on a stretched string?

Wave speed is the constant rate at which the disturbance moves horizontally along the string. Particle speed is the varying velocity of a specific point on the string as it moves vertically (transversely) up and down.

Why does a thicker string (of the same material) typically have a slower wave speed than a thinner string under the same tension?

A thicker string has a higher linear mass density ($\mu$), meaning it has more inertia per unit length. Because $v = \sqrt{\tau / \mu}$, the increased inertia makes it harder for the tension to accelerate the string segments, resulting in a slower propagation speed.

What common error is made when calculating linear mass density ($\mu$)?

A common error is using the total mass of the string ($m$) instead of the mass per unit length ($m/L$). Additionally, failing to convert units to the SI standard (kg/m) will result in an incorrect numerical value for speed.

What happens to the wave speed if the tension in a string is increased by a factor of four?

The wave speed will double. Since $v$ is proportional to the square root of tension ($\sqrt{\tau}$), increasing the tension by a factor of 4 results in $\sqrt{4} = 2$ times the original speed.

Why is the linear mass density ($\mu$) considered the 'inertial' property in the wave speed formula?

Inertia is the resistance of an object to changes in its motion. In the context of a string, $\mu$ represents how much mass must be moved for every meter the wave travels; more mass requires more force to achieve the same acceleration.

How does the wave speed change if you move from a heavy steel wire to a light nylon string, assuming tension is kept constant?

The wave speed will increase. Nylon has a much lower linear mass density ($\mu$) than steel; with less inertia to overcome, the same tension force can propagate the wave much faster.

What is the SI unit for linear mass density, and why is it important?

The SI unit is kilograms per meter (kg/m). It is essential for ensuring that the resulting wave speed is in meters per second (m/s) when tension is measured in Newtons ($kg \cdot m/s^2$).

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5. Waves & Particle Nature of Light

Wave Speed on a Stretched String

Summary

The speed of a mechanical wave on a stretched string is determined solely by the physical properties of the medium—specifically its tension and linear mass density. This relationship illustrates the fundamental principle that wave speed is a balance between the restoring force that returns the medium to equilibrium and the inertia that resists motion.

1. Definition & Core Concepts

Wave Speed ( $v$ ): The velocity at which the wave disturbance travels along the string. It is a property of the medium itself and remains constant regardless of the wave's frequency or amplitude.
Tension ( $\tau$ ): The pulling force exerted by the string, acting as the restoring force. Higher tension allows the string to return to its equilibrium position more quickly, increasing wave speed.
Linear Mass Density ( $\mu$ ): The mass per unit length of the string, defined as $\mu = m/L$ . This represents the inertia of the medium; a heavier string resists acceleration, thereby slowing the wave down.

Diagram of a transverse wave on a string showing the tension force vector and the wave velocity vector.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

The 'Frequency Fallacy': Many students incorrectly believe that shaking a string faster (increasing frequency) increases the wave speed. In reality, increasing frequency only shortens the wavelength; the speed remains fixed by the tension and density.
Mass vs. Linear Density: Do not use the total mass ( $m$ ) in the denominator of the speed formula. You must use the mass per unit length ( $\mu$ ).
Tension vs. Extension: Tension is a force (Newtons), not the distance the string is stretched. While stretching a string usually increases tension, the formula requires the force value.