# 2.10 Wave Speed on a Stretched Spring

## Wave Speed on a Stretched String

• The speed of a wave travelling along a string with two fixed ends is given by: • Where:
• T = tension in the string (N)
• μ = mass per unit length of the string (kg m–1)

• At the fundamental frequency, f0 of a stationary wave of length L, the wavelength, λ = 2L
• Therefore, according to the wave equation, the speed of the stationary wave is:

v = = f × 2L

• Combining these two equations leads to the equation for the fundamental frequency (sometimes referred to as the first harmonic): • Where:
• f = frequency (Hz)
• L = the length of the string (m)
• = the tension in the string (N)
• µ = mass per unit length (kg m-1)

• Mass per unit length, µ can be calculated by dividing the mass of the string by the length of the string  Diagram showing the first three modes of vibration of a stretched string with corresponding frequencies

#### Worked example

A guitar string of mass 3.2 g and length 90 cm is fixed onto a guitar.

The string is tightened to a tension of 65 N between two bridges at a distance of 75 cm. Calculate the

a) speed of the waves on the string

b) fundamental frequency of the string

Part (a)

Step 1: Write the known quantities in S.I. units

• Tension, T = 65 N
• Mass, m = 3.2 g = 3.2 × 10−3 kg
• Length of string, L = 90 cm = 0.90 m
• Mass per unit length, μ = 3.56 × 10−3 kg m−1

Step 2: Write the equation for speed on a string and calculate

• v = = f × 2L AND • So, v =  = 135

Step 3: Write the answer to the correct significant figures and include units

• The speed of the wave on the string, v = 140 m s−1

Part (b)

Step 1: Write the known quantities in S.I. units

• Tension, T = 65 N
• Length of string under tension, L = 75 cm = 0.75 m
• Mass per unit length, μ = 3.56 × 10−3 kg m−1 (from part (a))

Step 2: Identify the length of one wavelength at the fundamental frequency, f0 Step 3: Write the equation for fundamental frequency and calculate = 90.1

Step 3: Write the answer to the correct significant figures and include units

• The fundamental frequency, f0 = 90 Hz

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