The most soaring vocal melody is in Johann Sebastian Bach's...

The most soaring vocal melody is in Johann Sebastian Bach's Mass in B Minor. A portion of the score for the Credo section, number 9, bars 25 to 33, appears in Figure Pl7.21. The repeating syllable O in the phrase "resurrectionem mortuorum" (the resurrection of the dead) is seamlessly passed from basses to tenors to altos to first sopranos, like a baton in a relay. Each voice carries the foreground melody up through a rising passage encompassing an octave or more. Together the voices carry it from D below middle $\mathrm{C}$ to A above a tenor's high C. In concert pitch, these notes are now assigned frequencies of 146.8 $\mathrm{Hz}$ and 880.0 $\mathrm{Hz}$ . (a) Find the wavelengths of the initial and final notes. (b) Assume the chorus sings the melody with a uniform sound level of 75.0 dB. Find the pressure amplitudes of the initial and final notes. (c) Find the displacement amplitudes of the initial and final notes. (d) What If? In Bach's time, before the invention of the tuning fork, frequencies were assigned to notes as a matter of immediate local convenience. Assume the rising melody was sung starting from 134.3 $\mathrm{Hz}$ and ending at 804.9 $\mathrm{Hz}$ . How would the answers to parts (a) through (c) change?

Key Concepts

Frequency and Wavelength Relationship

This concept involves the fundamental relationship between the frequency of a sound wave and its wavelength, determined by the speed of sound in the medium (commonly air). The formula ? = v/f shows that for a constant speed, an increase in frequency results in a decrease in wavelength, and vice versa. This relationship is a cornerstone of acoustics and wave physics and is used to connect the spectral properties of a sound to its spatial characteristics.

Sound Pressure Level and Pressure Amplitude

Sound pressure level (SPL) measured in decibels (dB) is a logarithmic measure of the effective pressure of a sound relative to a reference pressure, usually 20 ?Pa in air. Converting SPL to a pressure amplitude involves inverting the logarithmic relationship, allowing one to quantify the pressure fluctuations in the medium that correspond to perceptible sound intensities. This conversion is critical in understanding how sound energy is delivered and measured in physical terms.

Displacement Amplitude in Acoustic Waves

Displacement amplitude refers to the maximum distance that air particles move from their equilibrium positions as a sound wave passes through. This concept is linked to the pressure amplitude and is calculated using the acoustic impedance of the medium along with the angular frequency. Understanding displacement amplitude is essential for connecting the measurable pressure variations of a sound to the actual motion of particles, providing insight into the energy and dynamics of the wave.

Acoustic Impedance

Acoustic impedance is a property of a medium that characterizes how much resistance it provides to the transmission of sound. It is typically defined as the product of the medium's density and the speed of sound. This parameter is crucial in relating pressure amplitude to displacement amplitude, as it quantifies the efficiency with which pressure variations in the wave cause the particles in the medium to move.

Transcript

00:01 So firstly, we'll calculate the wavelength lambda for the low node.

00:08 Lambda l is equal to the speed of sound in a, v, over the frequency of the node f.

00:17 The speed of sound is 343 meters per second, and the frequency of the note is 146 .8 hertz, or 146 .8 per second.

00:33 And this gives us the wavelength of the no note to be 2 .34 meters.

00:43 The wavelength of the high note we get from the same formula v over f with a frequency now changing.

00:52 So the speed of the wave stays unchanged, it's the speed of sound in a but the frequency of the high node is 880 hertz or 880 a second.

01:05 Hence we get the wavelength of the high note to be 0 .3 .9 meters.

01:19 Next for part b, we are given the sound level, and the sound level we know is defined as 10 decibels multiplied by the log of the intensity of a sound i, which we wish to find, divided by the reference intensity 10 to the minus 12.

01:47 Watts per square meter and we're given the sound level to be 75 decibels so if we rearrange this equation we can find the intensity i and if we rearrange us we see that i is equal to 3 .16 times 10 the minus 5 watts a square meter now that we have i we can calculate the pressure amplitude since we know that i is defined as the pressure amplitude delta p max squared over two times the density of the medium in which the wave travels times the speed of the wave so if you rearrange this we can find the pressure amplitude as required so the delta p max is simply the square root of i times 2 row v so that's the square root we just calculated the intensity to be 3 .16 times 10 to the minus 5 watts per square meter so it's addressed the units multiplied by 2 and the density of a which is 1 .2 kg per cubic meter and the speed of sound in a 3003 meters per second during the calculation we get the pressure amplitude to be 0 .161 pass curve and note that this pressure amplitude is for both the low and the high notes...

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