Waves On A String Answer Key

Waves on a String: A Comprehensive Guide with Answers

Understanding wave phenomena is crucial in physics, and waves on a string provide a readily accessible model to explore fundamental concepts. This comprehensive guide delves into the physics of waves on a string, covering key concepts, calculations, and providing detailed answers to common problems. We'll explore everything from basic definitions to more advanced topics like standing waves and resonance.

What are Waves on a String?

Waves on a string, also known as transverse waves, are disturbances that propagate along a stretched string or similar medium. The disturbance is perpendicular to the direction of wave propagation. Imagine plucking a guitar string – the string moves up and down (transverse motion), but the wave travels along the length of the string.

Key Terminology:

• Amplitude (A): The maximum displacement of the string from its equilibrium position. A larger amplitude means a more energetic wave.
• Wavelength (λ): The distance between two consecutive points on the wave that are in the same phase (e.g., two consecutive crests or troughs).
• Frequency (f): The number of complete oscillations (cycles) the wave completes per unit time, typically measured in Hertz (Hz).
• Period (T): The time it takes for one complete oscillation to occur. It's the reciprocal of the frequency (T = 1/f).
• Wave Speed (v): The speed at which the wave travels along the string. It's related to the wavelength and frequency by the equation: v = fλ
• Tension (T): The force applied to the string, stretching it taut. Higher tension generally leads to faster wave speeds.
• Linear Mass Density (μ): The mass per unit length of the string. A thicker string has a higher linear mass density.

The Wave Equation: Connecting Tension, Mass Density, and Wave Speed

The speed of a wave on a string is fundamentally determined by the properties of the string itself: its tension and linear mass density. The relationship is expressed by the following equation:

v = √(T/μ)

Where:

• v is the wave speed
• T is the tension in the string
• μ is the linear mass density of the string

This equation highlights that increasing tension increases wave speed, while increasing linear mass density decreases wave speed.

Standing Waves and Resonance

When a string is fixed at both ends, specific frequencies of vibration can create standing waves. These are waves that appear stationary, with points of maximum displacement (antinodes) and points of zero displacement (nodes).

Conditions for Standing Waves:

Standing waves only form when the string length (L) is an integer multiple of half the wavelength:

L = n(λ/2) where n is an integer (1, 2, 3...)

This leads to the frequencies of the standing waves (harmonics or overtones):

fₙ = n(v/2L) where n is the harmonic number (1st harmonic, 2nd harmonic, etc.)

The first harmonic (n=1) is the fundamental frequency, the lowest frequency at which a standing wave can form. Subsequent harmonics are integer multiples of the fundamental frequency.

Solving Problems: Worked Examples

Let's tackle some example problems to solidify our understanding:

Problem 1: A string with a linear mass density of 0.01 kg/m is stretched with a tension of 100 N. What is the speed of a wave traveling along this string?

Answer:

We use the wave speed equation: v = √(T/μ)

Substituting the given values: v = √(100 N / 0.01 kg/m) = √(10000 m²/s²) = 100 m/s

Therefore, the speed of the wave is 100 m/s.

Problem 2: A string of length 1 meter is fixed at both ends. If the wave speed on the string is 200 m/s, what are the frequencies of the first three harmonics?

Answer:

The frequency of the nth harmonic is given by: fₙ = n(v/2L)

• 1st harmonic (n=1): f₁ = 1 * (200 m/s) / (2 * 1 m) = 100 Hz
• 2nd harmonic (n=2): f₂ = 2 * (200 m/s) / (2 * 1 m) = 200 Hz
• 3rd harmonic (n=3): f₃ = 3 * (200 m/s) / (2 * 1 m) = 300 Hz

The frequencies of the first three harmonics are 100 Hz, 200 Hz, and 300 Hz.

Problem 3: A guitar string has a fundamental frequency of 440 Hz (A4 note). If the string's length is 0.65 meters and the wave speed is 300 m/s, what is the wavelength of the fundamental frequency?

Answer:

We know that for the fundamental frequency (n=1): L = λ/2

Therefore, λ = 2L = 2 * 0.65 m = 1.3 meters

The wavelength of the fundamental frequency is 1.3 meters.

Advanced Concepts: Damping and Dispersion

While the simplified model above provides a good foundation, real-world strings exhibit damping and dispersion.

Damping: Energy is lost from the wave due to friction within the string and with the surrounding air. This causes the amplitude of the wave to decrease over time.

Dispersion: The wave speed depends on the frequency of the wave. This means different frequency components of a complex wave will travel at different speeds, leading to the wave spreading out or distorting over time.

Applications of Waves on a String:

Understanding waves on a string has numerous applications, including:

• Musical Instruments: Guitars, violins, and pianos all rely on vibrating strings to produce sound.
• Telecommunications: Signals are transmitted along cables using waves.
• Seismic Studies: The principles of wave propagation are used to understand seismic waves.

Conclusion:

Waves on a string offer a fundamental and accessible model for studying wave phenomena. By understanding the key concepts like wavelength, frequency, wave speed, tension, and linear mass density, and their relationships, we can analyze and predict the behavior of waves in various scenarios. This guide provided a comprehensive overview of the topic, including worked examples to further enhance your understanding. Remember that while simplified models are useful, real-world scenarios often incorporate factors like damping and dispersion, which should be considered for more accurate analysis. This foundational knowledge will serve as a springboard for exploring more complex wave phenomena in physics.