Understanding Wave Velocity on a String

Wave Velocity on a String is a fundamental concept in JEE Main Physics. Understanding how wave speed varies with string properties and tension is essential for solving advanced wave problems. Mastery of this topic also strengthens conceptual knowledge for questions linking wave motion to oscillatory systems.

Wave Velocity on a String: Definition and Physical Meaning

Wave velocity on a string refers to the rate at which a disturbance, such as a pulse or vibration, travels along a stretched string. It is a type of transverse wave, where particles of the string oscillate perpendicular to the wave’s direction of travel.

The ability to calculate wave speed allows you to solve questions involving vibrating strings, musical instruments, and resonance. It is a key application of wave mechanics in the JEE Main syllabus.

Key Equations Governing Wave Speed in Strings

The speed of a transverse wave along a stretched string depends on two main factors: tension and linear mass density. This is described by the universal wave equation:

Let F_T represent the tension in the string, and μ the linear mass density (mass per unit length):

Wave speed, v
= \( \sqrt{\frac{F_T}{\mu}} \)

Here, increasing the tension F_T raises the wave speed, while increasing the linear density μ lowers it.

Properties Influencing Wave Velocity on a String

Several parameters define how fast waves can travel along a given string. For conceptual clarity, always consider:

• Tension in the string, set by any hanging weights or fixed supports
• Linear mass density, calculated as mass divided by length
• Material characteristics, as different substances yield different densities
• String length, which determines allowed standing wave patterns but not wave speed directly
• External factors such as temperature, affecting tension or elasticity

For more on wave types, refer to our comprehensive Longitudinal And Transverse Waves guide.

Derivation of the Wave Velocity Formula for JEE Main

Step 1: Consider a string stretched under tension F_T. Take a small segment of the string of length Δx, mass μ Δx.

Step 2: Analyzing the vertical forces and acceleration on this segment, apply Newton’s second law:

Net force = mass × acceleration
= μ Δx × ∂²y/∂t²

Step 3: Using small angle approximation, the net vertical force due to the tension is:

F_net = F_T [ ( ∂y/∂x )_x+Δx - ( ∂y/∂x )_x ]

Step 4: Equate and divide both sides by Δx, then take Δx → 0:

F_T ∂²y / ∂x² = μ ∂²y / ∂t²

Step 5: Rearranging yields the standard wave equation for a vibrating string:

( ∂²y / ∂x² ) = (1/v² ) ∂²y / ∂t²

Comparing gives v = \( \sqrt{F_T/\mu} \) as the final expression for string wave speed.

Examples and Applications: JEE Main Context

A guitar string with tension 100 N and linear density 0.01 kg/m:

v = \( \sqrt{ \frac{100}{0.01} } \)
= \( \sqrt{10\,000} \)
= 100 m/s

Instruments and laboratory demonstrations often use this equation to set desired frequencies by controlling tension or string thickness.

To review practical wave phenomena, see Oscillations And Waves for worked examples relevant for JEE Main.

Table: Key Parameters Affecting Wave Velocity on a String

Common Mistakes and Tips

JEE Main aspirants often confuse frequency and wave speed. Wave velocity relies only on string tension and density—not on the source frequency or string length. Remember to always use SI units for calculations, as required for JEE Main Physics.

For more advanced concepts like beats, explore Beat Frequency Formula, or apply wave velocity concepts to communication devices in our Communication Systems resource.

Mastering the principles of string wave speed gives you an edge in solving complex questions on resonance, harmonics, and lab-based measurement tasks, all of which are important for your JEE Main preparation. For reliable study material and expert problem-solving strategies, Vedantu remains trusted by top JEE scorers nationwide.