Question

A \(15.0-\mathrm{cm}\) violin string, fixed at both ends, is vibrating in its \(n=1\) mode. The speed of waves in this wire is \(250 \mathrm{~m} / \mathrm{s}\), and the speed of sound in air is \(348 \mathrm{~m} / \mathrm{s}\). What are \((a)\) the frequency and \((b)\) the wavelength of the emitted sound wave?

Short answer

The frequency of the emitted sound wave is 1666.67 Hz and the wavelength is 0.2088 m.

Step-by-step solution

Formula for frequency

The frequency of the vibrating string can be found using the formula for the speed of a wave on a string, which is \(v = f \lambda\), where \(v\) is the speed of the wave, \(f\) is the frequency, and \(\lambda\) is the wavelength. Given that the string is vibrating in its \(n=1\) mode, it means that the length of the string represents one full wavelength. Thus, \(\lambda = 0.15m\). Solving for \(f\), we get \(f = \frac{v}{\lambda}\).

Calculate frequency

We substitute the given values into the formula from step 1: \(f = \frac{250ms^{-1}}{0.15m} = 1666.67 Hz\). So this is the frequency of the vibrating string.

Formula for wavelength of sound wave

The wavelength of the emitted sound wave can be found using the formula for the speed of sound, which is \(v = f \lambda\), where \(v\) is the speed of sound, \(f\) is the frequency, and \(\lambda\) is the wavelength. Solving for \(\lambda\), we get \(\lambda = \frac{v}{f}\).

Calculate the wavelength

We substitute the given values and frequency calculated into the formula from step 3: \(\lambda = \frac{348ms^{-1}}{1666.67 Hz} = 0.2088 m\). So this is the wavelength of the emitted sound wave.

Key concepts

Wave Speed

Understanding the speed at which waves travel is fundamental in solving problems regarding vibrating strings. Wave speed is the distance a wave travels per unit time and is often denoted by the symbol 'v'. In the context of a stringed musical instrument like a violin, wave speed refers to how quickly disturbances, caused by plucking or bowing the string, propagate along the length of the string.

The medium through which the wave is traveling substantially affects this speed. For example, tension and the mass per unit length of the string are crucial factors for wave speed in a string. The higher the tension and the lower the mass per unit length, the faster the wave will travel. This concept is pivotal, as knowing the wave speed is the starting point for calculating both the frequency and wavelength of sound waves produced by the vibrating string.

Sound Wavelength

The wavelength is the physical distance between successive crests (or troughs) of a wave, denoted as \(\lambda\). In the world of acoustics, the wavelength of a sound wave determines its pitch; shorter wavelengths correspond to higher frequencies (higher pitches), while longer wavelengths correspond to lower frequencies (lower pitches).

For a string instrument, when we say the string is vibrating at a certain frequency in its fundamental mode (also known as the first harmonic), the wavelength of the vibration is twice the length of the string. Sound wavelength in the air is then considered a transposition of these vibrations, as the air particles affected by the string's movement propagate the sound away from the instrument. This transference is why knowing the speed of sound in air (348 m/s in the given problem) allows us to calculate the wavelength of that sound and understand the pitch of the note being played.

Harmonic Modes

Harmonic modes, or simply harmonics, are specific patterns at which a string can naturally resonate or vibrate when disturbed. Each mode corresponds to a standing wave pattern characterized by nodes, points of no displacement, and antinodes, points of maximum displacement.

The first mode, also known as the fundamental frequency, is the lowest frequency at which a string vibrates. It has no internal nodes and the string vibrates as a whole, with nodes only at the fixed ends. Higher modes, known as overtones or harmonics, have additional internal nodes. The frequency of each mode is an integral multiple of the fundamental frequency, significantly shaping the timbre and the overall sound quality of a musical instrument. When solving problems involving vibrating strings, identifying the harmonic mode is key to determining the correct wavelength for calculations.

Frequency Calculation

Frequency calculation in the context of vibrating strings involves determining how often the string completes a vibration cycle per second. The frequency, measured in hertz (Hz), is calculated by dividing the wave speed by the wavelength (\(f = \frac{v}{\lambda}\)). This relationship helps us understand that for a string of a certain length and tension, the frequency of vibration is dependent on mode.

In our exercise, the frequency of the string and the emitted sound wave are inherently linked. Once we have calculated the frequency of the string vibrating in its first mode (fundamental frequency), we can use the speed of sound in air to find the wavelength of the sound wave heard. This step is indispensable for musicians and instrument makers, as it directly affects the pitch of the notes produced by stringed instruments.