The guide to musical instruments
By Zimo Wang, 22nd May 2022
Translated by Yifan Wang, 21st Oct 2023
For most of us, music is a very important part of our lives, and most of this music is produced by musical instruments. As we all know, sound is transmitted to the ears through sound waves, as waves are a means of transferring energy. When we listen to music, the refreshing feeling we get in our ears is also a result of the conservation of energy. So today, let's dive in and take a look at how to play these musical instruments! Instruments are mainly divided into string instruments and wind instruments, with percussion instruments also being part of the mix, but we will primarily focus on the playing techniques of the former two. Part 1 of this guide will introduce string instruments, and part 2 will cover wind instruments. Without further ado, let's get started!
If you don't want to delve into the series of formula derivations and physics content, you can skip the end of part 1, where you'll find all the relevant conclusions about these instruments.
PART 1: String Music
When it comes to string instruments, our primary focus should be on the "strings" themselves. So, in this section, we will be analyzing the wave propagation present in string instruments.
1.1 The Wave and Sine Functions
Firstly, we need to list the function of a wave (which as a side note, is completely different from Schrödinger's wave function!).
Figure 1- Graph of the function of a wave
In Figure 1, the black lines represent the coordinate axes, and the blue lines represent the propagation of a wave on a string. We assume that the initial expression for the string is f(x), and after a time t, the expression for this wave is f(x - vt), where v is the wave's propagation speed. This should be easy to understand as vt simply represents the amount by which the function shifts horizontally, and the "-" sign indicates leftward translation, all mathematical concepts we had learnt in middle school. Therefore, the function can be expressed as:
If you think that because the “-” sign means that the wave travels leftwards, the “+” sign means that the wave travels to the right, you’ll be absolutely correct. The addition or subtraction in the expression indicates whether the wave is moving to the left or to the right.
However, upon further consideration, you'll realize that this formula cannot be quantitatively calculated because you don't know the exact function f(x). Therefore, we make an assumption that all the waves we are calculating are sinusoidal curves, rather than the type of wave shown in Figure 1 with only one peak.
Figure 2-Sine Wave
As seen from above, figure 2 represents a sinusoidal wave. You might also be wondering, “Why not use a cosine wave?” In actuality, both can be used, but the sine wave is more convenient because it involves one less negative sign when taking the first derivative. Moreover, according to trigonometric identities, sine and cosine waves can be converted into each other.
In Figure 2, 'A' stands for amplitude, and 'λ' stands for wavelength. So, let's now explore what the earlier mentioned function f(x) is. It is certainly related to the sine function, but it's not just f(x) = sin(x). The function has been vertically stretched by 'A' units, so we assume:
Additionally, the presence of 'm' within the equation is because we cannot guarantee that the period of each wave is always 2π. This is related to the concepts of period and frequency, so we need to multiply the sine function by an unknown variable 'm' to account for the horizontal stretching or compression of the wave. Next, we can determine 'm' when x equals 'λ,' which represents one complete period. Because when x = λ, f(x) = 0, so:
As a single period is 2π, we can obtain the relation m=2π/λ. With that information, we now know the specific function for f(x). Subsequently, when we substitute f(x) back into the original wave function we started with, we can obtain:
At this point, the derivation of the sinusoidal wave function can essentially be concluded. However, as the 'v' in the formula above can be somewhat awkward to measure precisely, we can simply avoid measuring ‘v’ by substituting it with λ/T (wavelength divided by the period). Thus, by substituting, we eventually obtain:
1.2 Derivation of velocity on a string
As mentioned earlier, measuring 'v' can be challenging. So, how should we calculate the speed of propagation on the string? Yep, you guessed it, through some "straightforward" derivatives.
Figure 3-Calculate the magnitude of the propagation velocity
For example, we will use Figure 3, the illustration shown above. Firstly, let's assume there is a brown string, and that the blue segment Δs represents a very small part of it, with the blue segment also having a curvature radius of R. The segment Δs is subject to equal tension T at both ends of the string. The horizontal components of T cancel each other out, while the vertical component acts as the centripetal force. As when θ is sufficiently small, sin(θ) approaches θ, we can thus derive the following equation:
Next up, we need to introduce a quantity called μ, which represents the mass per unit length of the string (unit is kg/m). Using μ, we can then arrive at this equation for mass:
Finally, substituting the formula for centripetal force, we get:
So, this is the formula for the speed of propagation on the string, which quite surprisingly is only related to tension and linear density! With this formula, we can also derive quite a useful application. For example, when you're tuning a guitar, you may notice that tightening of the strings results in a higher pitch. This is because when the strings are tighter, the tension T is greater, and the speed v is higher. With a constant wavelength λ, the frequency f naturally increases. Since a higher frequency corresponds to a higher pitch, the sound is naturally higher. The same principle applies to thicker strings; you just need to account for the change in linear density μ. We will revisit this in the next section, and by then, you'll be able to perform quantitative calculations!
1.3 Calibrating your performance
When two identical waves (with the same frequency and wavelength) propagate in opposite directions, one to the right and one to the left, their superposition creates the following function:
These waves, which are identical and propagate in opposite directions, when superimposed, create what we call a standing wave, and its equation is the third one seen above. By observing the standing wave, you can see that there are certain points that remain stationary, while there are other points that have a consistent maximum amplitude. So, how do we calculate those stationary points? It's simply a matter of substituting the function for Σy, and as we know that it has zero variation with respect to time 't’, the part in front of the cosine function in the Σy equation should be equal to zero, or else its 'y' value wouldn't remain at zero. Hence, we have:
We refer to these stationary points as nodes. And the points where the amplitude is consistently at its maximum can also be determined. Since the maximum value of sin is ±1, to achieve maximum amplitude, we need to set sin to ±1. Therefore:
We call these points with maximum displacement antinodes. Wondering why we need to know these points? Don't worry, we'll soon apply this knowledge to stringed musical instruments!
Figure 4-Harmonic waves
We'll assume that the string in the diagram is our stringed instrument's string. Since the string is tightened at both ends of the instrument, the endpoints of the string can only be nodes and not antinodes. Therefore, what's depicted in Figure 4a represents the case with the longest wavelength and the lowest frequency, which we can call the fundamental frequency (or the first harmonic). Figure 4b shows another standing wave, but its wavelength is half that of the fundamental frequency, which we can refer to as the second harmonic. Figure 4c represents the third harmonic. These harmonics have regular wavelengths and can be described by the following formula:
In the formula, L represents the length of the string, and 'n' indicates which order the harmonic it is. For example, 'λ' is the wavelength of the fundamental frequency (first harmonic), and if you substitute 'n = 1', you get that its wavelength is twice the length of the string. So now, when you measure the length of your stringed instrument's string and multiply it by 2, you'll obtain the wavelength of its fundamental frequency. Additionally, when you’re inevitably going to buy a string instrument, don't ask about the length of the string; instead ask, "Hello, could you please tell me the wavelength of the fundamental frequency (or first harmonic) of this string?" If they can provide that information, you can divide that number by 2 to find the string's length. But most likely, they won't be able to answer, so you can share this article with them.
Back to the topic at hand, just knowing the wavelength is not particularly useful, but instead, it's the frequency that matters. We know that frequency is equal to velocity divided by wavelength, so we can easily determine the frequency 'f':
Do you still remember that in the second chapter, we derived a formula for the linear velocity? We can also substitute that to obtain the formula:
Of course, this formula has many practical applications. For example, while some instruments like the guitar have frets to guide you, many instruments such as the violin require you to find the exact positions for each note. How can you accurately determine the location of a specific note? Calculations can be quite useful in this regard.
Figure 5-Problem of a violin
The challenge you're presenting involves calculating the position to move to in order to reach the desired note, which is slightly higher than middle C. Assuming you know the position of middle C, represented as 'a' cm away from the nut, and you want to find how much to move to reach the desired D note, which is 'b' cm away from middle C, you can indeed use the formula you mentioned to calculate 'a - b' cm. Here's how you can do it:
Then you can proceed to calculate the quantity you're looking for, which is 'a - b' cm:
So, you only need to measure the distance 'a' from middle C to the nut, and then move your hand to the right by (0.11a) cm to reach the desired D note. You can give it a try! This concept applies to other musical instruments as well.
1.5 Conclusion
Well, that was a long read, wasn’t it? So, let's summarize today's content. We started by learning the fundamental sin function in music and then derived the velocity on a string. Next, we explored the harmonic formulas, which can help you have more academic discussions when buying instrument strings. For instance, you can ask others about the linear density of a string or inquire about the wavelength of the first harmonic of a specific string. This will certainly help you articulate your thoughts more clearly. Lastly, when you can't find the right pitch, don't forget to use the calibration method mentioned earlier!
PART 2: Wind Music
2.1 Function of a sound wave
When it comes to sound waves, the most significant difference from the waves we discussed last time is that sound waves are longitudinal waves. From a microscopic perspective, longitudinal waves have no amplitude, as shown in Figure 6.
Figure 6-(Up) Longitudinal wave; (Down) Transverse wave
As for the fact that in longitudinal waves, the direction of particle motion is parallel to the direction of wave propagation, and there is no vertical wave amplitude, we need a different function to represent such waves:
Here, 'Zmax' replaces the position of the wave amplitude. 'Zmax' refers to the maximum distance a particle moves in that direction. It's similar to the maximum vertical displacement of the wave amplitude, just with a different letter and direction. For sound waves, we often also need to know something else as well, that being pressure, 'P':
Where ΔPmax is the maximum change in pressure. When you think about it, you'll realize that ΔPmax should occur at the point Zmax. This is because when particles move the maximum distance horizontally, they become tightly packed, leading to higher pressure, whilst when particles are at their initial positions, the pressure should be minimal. This is a conclusion we can roughly deduce based on common knowledge.
But how can we scientifically prove this? Pressure and Zmax are certainly related, so we can consider this connection and ultimately arrive at the following solution. Before that, let's take a look at an illustrative diagram to prepare for the proof.
Figure 7-Schematic of the derivation of the formula for pressure and position
In Figure 7a, Δx stays the same as the stuff discussed in part 1 of this article, representing a small length. After a short time 't', Figure 7a transforms into Figure 7b. Firstly, you can see that the entire wave has moved a bit to the right; A becomes A', and B also shifts to B'. If you merely glance at the diagram, you'll notice that AB equals A'B'. However, in actuality, they are not equal. The reason behind this is the influence of pressure differences. So, we can introduce another quantity, Δz, which equals AB - A'B', representing the length change due to pressure, which also equals z1 - z2. Now, let's begin the derivation; it's actually quite simple. According to the definition bulk modulus (B), we have:
Assuming that the cross-sectional area of the wave in Figure 7 is 'A', we can calculate the volume using V = A * some length:
Taking the limit and obtaining a partial derivative:
Using the function with respect to 'z' is given in the first section of this article, you can then substitute it and then differentiate:
And since the function of ΔP perfectly cancels out the sine part on the right side of the equation above, you can ultimately obtain:
Hence, the final proof shows that for longitudinal waves, when particle displacement reaches its maximum, pressure also reaches its maximum.
2.2 Resonance
Figure 8-Resonance and standing wave formation in an empty tube
From the diagram, we can see a structure almost identical to the one seen in the previous article, with several different harmonics. However, this time, they are confined within the two open ends of the tube rather than staying on a string. A tube with two open ends is similar in structure to wind instruments. In such a structure, both ends of the tube are actually antinodes, which are the points of maximum displacement, as there are no constraints here. But theoretically, it's not a pure antinode, although we won't delve into that here.
At the top is the first harmonic, which is the fundamental frequency inside the tube. Its wavelength is twice the length of the tube. In the second harmonic, the wavelength is equal to the tube's length, and in the third harmonic, the wavelength is two-thirds of the tube's length. Therefore, we can write down the pattern and deduce:
Where 'n' represents which harmonic the wave is, and if you recall the wavelength formula from the previous article, you'll find that it's exactly the same. We can continue to derive and calculate the frequency. All you need to do is divide by the speed.
However, are tubes with two open ends exactly the same as waves on a string? Certainly not. Aside from the different types of waves and velocity calculations mentioned earlier, from the perspective of standing waves, the ends of the string are nodes because they are fixed, whereas the ends of the tube are antinodes because they are open.
Finally, we can apply the velocity formula and the subsequent steps, which are similar to the content from the previous article, thus there's no need to go into further detail here.
That concludes our two-part exploration of waves. Learning about sound through musical instruments is definitely an interesting approach. Of course, there are many other fascinating topics to delve into, such as considerations in instrument making and Fourier transforms, so we'll save those for another time!
Thanks for reading!