The frequency of vibrations of a vibrating violin string is...

The frequency of vibrations of a vibrating violin string is given by $$ f=\frac{1}{2 L} \sqrt{\frac{T}{\rho}} $$ where $L$ is the length of the string, $T$ is its tension, and $\rho$ is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA, 2002).] (a) Find the rate of change of the frequency with respect to (i) the length (when $T$ and $\rho$ are constant), (ii) the tension (when $L$ and $\rho$ are constant), and (iii) the linear density (when $L$ and $T$ are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency $f .$ (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates, (ii) when the tension is increased by turning a tuning peg, (iii) when the linear density is increased by switching to another string.

The frequency of vibrations of a vibrating violin string is given by
$$
f=\frac{1}{2 L} \sqrt{\frac{T}{\rho}}
$$
where $L$ is the length of the string, $T$ is its tension, and $\rho$ is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA, 2002).]
(a) Find the rate of change of the frequency with respect to
(i) the length (when $T$ and $\rho$ are constant),
(ii) the tension (when $L$ and $\rho$ are constant), and
(iii) the linear density (when $L$ and $T$ are constant).
(b) The pitch of a note (how high or low the note sounds) is determined by the frequency $f .$ (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note
(i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates,
(ii) when the tension is increased by turning a tuning peg,
(iii) when the linear density is increased by switching to another string.

Key Concepts

Vibrating String Frequency Formula

This concept involves the mathematical model that describes the frequency of a vibrating string based on its physical parameters. The formula f = (1/(2L))?(T/?) encapsulates how the frequency depends on the string's length, the tension applied, and the linear density, showcasing the underlying physics of wave propagation on a stretched string.

Partial Differentiation

This is the process of finding the derivative of a multivariable function with respect to one variable while holding the other variables constant. It is used to determine how a small change in one specific parameter (length, tension, or density) impacts the frequency of the string's vibration, independently of the other parameters.

Interpretation of Derivative Signs

This concept relates to understanding the physical meaning behind the positive or negative values of the computed derivatives. A positive derivative indicates that an increase in the parameter will lead to an increase in frequency (and thus a higher pitch), while a negative derivative indicates that an increase in the parameter results in a decrease in frequency (and a lower pitch).

Parameter Sensitivity in Vibrating Strings

This key idea focuses on how adjustments in the physical properties of a string—specifically its length, tension, and linear density—affect its vibrational frequency and, consequently, its musical pitch. This understanding is crucial in musical acoustics, as it explains why shortening a string raises the pitch, increasing tension raises the pitch, and increasing density lowers the pitch.

Transcript

00:01 So, given in the question, the frequency of vibration of a vibrating string is given as f that is equals to 1 by 2l under root t by rho, where l is the length of the string, t is the tension in the string and rho is the linear density of the string.

00:37 Now, it has been asked to find the rate of change of the frequency of vibration as a function of the length of the string, keeping the tension and linear density constant.

00:59 In the second part, it is asked to find the rate of change of frequency with respect to tension, keeping l and rho constant.

01:10 And in the third part is to find the rate of change of frequency as a function of linear density, keeping l and t constant.

01:20 In part b, it has been asked that what happens to the pitch when in the first case, the effective length of the string is decreased.

01:32 In the second case, when the tension in the string is increased.

01:40 And in the third case, the linear density of the string is increased.

01:54 So, solving the first part, part a and the first part of part a.

02:01 So, taking the derivative of f with respect to l keeping t and rho constant.

02:06 So, we can take out t and rho out of the derivatives.

02:12 So, it is 1 by 2 root over t by rho d by dl of 1 by l.

02:19 So, d by dl of 1 by l is minus 1 by l square.

02:25 So, it would be 1 by 2 root t by rho times minus 1 by l square.

02:32 Finally, we are getting df by dl to be minus 1 by 2 l square under root t by rho.

02:41 Now, in the second is to find the rate of change of f with respect to t, keeping l and rho constant.

02:49 So, we can take out l and rho out of the derivative sign and find the derivative of root t with respect to t.

02:59 That value is 1 by 2 root t...

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