Standing Waves (Nodes at Both Ends)

Introduction to Wave Superposition

• Superposition Principle: When two or more waves overlap in the same region of space, the resultant displacement at any point is the vector sum of the displacements of the individual waves.
• Constructive Interference: Occurs when waves are in phase (crests align with crests, troughs with troughs), resulting in a larger amplitude.
• Destructive Interference: Occurs when waves are out of phase (crests align with troughs), resulting in a smaller amplitude or cancellation.

KEY TAKEAWAY: Wave superposition is the fundamental principle behind the formation of standing waves.

Formation of Standing Waves

• Standing waves, also known as stationary waves, are formed by the superposition of two identical waves traveling in opposite directions.
• In the context of VCE Physics, we primarily focus on standing waves formed on strings or in air columns with fixed ends or nodes at both ends.

Conditions for Standing Wave Formation

1. Wave Reflection: A wave travels along a medium (e.g., a string) and is reflected at a fixed end. The reflected wave is inverted (180° phase shift).
2. Interference: The incident wave and the reflected wave interfere.
3. Resonance: At specific frequencies, the interference pattern produces a stable standing wave pattern. These frequencies are called resonant frequencies or harmonics.

STUDY HINT: Visualize the waves traveling back and forth and interfering to create the standing wave pattern.

Nodes and Antinodes

• Nodes: Points along the medium where the displacement is always zero. They occur where the interfering waves are always 180° out of phase, causing complete destructive interference. For our case, the ends are always nodes.
• Antinodes: Points along the medium where the displacement has maximum amplitude. They occur where the interfering waves are in phase, causing constructive interference.

Harmonics (Modes of Vibration)

• Harmonics are the specific resonant frequencies at which standing waves are formed. Each harmonic corresponds to a different standing wave pattern.
• The first harmonic is also called the fundamental frequency ($f_1$).
• Higher harmonics are integer multiples of the fundamental frequency.

APPLICATION: Musical instruments rely on standing waves to produce sound. Different harmonics create different tones or timbres.

Mathematical Description of Standing Waves

Wavelength and Length Relationship

For a string of length $L$ fixed at both ends, the possible wavelengths ($\lambda_n$) for standing waves are given by:

$$L = n \frac{\lambda_n}{2}$$

where:

• $L$ is the length of the string.
• $n$ is an integer (1, 2, 3, …) representing the harmonic number.
• $\lambda_n$ is the wavelength of the $n$th harmonic.

Rearranging for $\lambda_n$:

$$\lambda_n = \frac{2L}{n}$$

Frequency and Velocity Relationship

The frequency ($f_n$) of the $n$th harmonic is related to the wave velocity ($v$) and wavelength ($\lambda_n$) by:

$$v = f_n \lambda_n$$

Substituting $\lambda_n = \frac{2L}{n}$:

$$f_n = \frac{nv}{2L}$$

where:

• $f_n$ is the frequency of the $n$th harmonic.
• $v$ is the wave velocity on the string.
• $L$ is the length of the string.
• $n$ is the harmonic number (1, 2, 3, …).

Fundamental Frequency

The fundamental frequency ($f_1$) is the lowest resonant frequency and corresponds to $n = 1$:

$$f_1 = \frac{v}{2L}$$

Higher Harmonics

The frequencies of higher harmonics are integer multiples of the fundamental frequency:

$$f_n = nf_1$$

Table of Harmonics

REMEMBER: L = n(lambda/2) - helps recall the length and wavelength relationship.

Wave Speed on a String

The speed of a transverse wave on a string is determined by the tension ($T$) in the string and the linear mass density ($\mu$) of the string:

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

where:

• $v$ is the wave speed.
• $T$ is the tension in the string (in Newtons).
• $\mu$ is the linear mass density (mass per unit length, in kg/m).

COMMON MISTAKE: Forgetting to convert units (e.g., cm to m, grams to kg) when calculating wave speed or frequency.

Examples and Applications

Example 1: Guitar String

A guitar string of length 0.65 m is fixed at both ends. The speed of the wave on the string is 195 m/s. Calculate the fundamental frequency and the frequency of the second harmonic.

• Fundamental frequency ($f_1$):

$$f_1 = \frac{v}{2L} = \frac{195}{2 \times 0.65} = 150 \text{ Hz}$$

• Second harmonic ($f_2$):

$$f_2 = 2f_1 = 2 \times 150 = 300 \text{ Hz}$$

Example 2: Determining Wave Speed

A string of length 1.5 m and mass 0.015 kg is fixed at both ends. The string is under a tension of 45 N. Calculate the speed of the wave on the string and the fundamental frequency.

• Linear mass density ($\mu$):

$$\mu = \frac{m}{L} = \frac{0.015}{1.5} = 0.01 \text{ kg/m}$$

• Wave speed ($v$):

$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{45}{0.01}} = \sqrt{4500} = 67.08 \text{ m/s}$$

• Fundamental frequency ($f_1$):

$$f_1 = \frac{v}{2L} = \frac{67.08}{2 \times 1.5} = 22.36 \text{ Hz}$$

EXAM TIP: Practice applying the formulas for wavelength, frequency, and wave speed in different scenarios. Pay attention to units and significant figures.

Factors Affecting Standing Wave Frequencies

1. Length of the string (L): Shorter strings produce higher frequencies. ($f \propto \frac{1}{L}$)
2. Tension in the string (T): Higher tension produces higher frequencies. ($f \propto \sqrt{T}$)
3. Linear mass density (μ): Lower linear mass density produces higher frequencies. ($f \propto \frac{1}{\sqrt{\mu}}$)

VCAA FOCUS: VCAA often tests the relationships between length, tension, linear mass density, and the resulting frequencies of standing waves. They might provide scenarios where one of these parameters is changed and ask how the frequency is affected.