Video: Understanding How the Density of a String Affects the Frequency and Wavelength of Waves Travelling along It

When a note on a piano is played, a hammer inside the piano strikes one of the strings of the piano, producing a standing wave on the string. Each of the following describes how the fundamental frequency and fundamental wavelength of the string may change when other properties of the string are changed.

06:24

Video Transcript

When a note on a piano is played, a hammer inside the piano strikes one of the strings of the piano, producing a standing wave on the string. Each of the following describes how the fundamental frequency and fundamental wavelength of the string may change when other properties of the string are changed.

Okay, before we get to these other descriptions, let’s consider what’s happening in this situation. We have a string on a piano. Let’s say that this is it. And we know it’s fastened at both ends. This string is set up to vibrate so that when it does, it produces a note. In order to make this happen, a device called a hammer strikes the string and produces a standing wave on it. In this context, a standing wave is one that doesn’t exhibit any horizontal motion left or right, but only vertical motion up and down.

As this string vibrates, we’re specifically interested in the fundamental frequency and fundamental wavelength of the string. The fundamental wavelength of a string is the longest wavelength that can fit on it. The two factors determining this wavelength are the endpoints of the string, which in this case are both fixed or fastened. And then, given that, the only other factor affecting the fundamental wavelength is the string length, what we can call capital 𝐿.

If this string were oscillating at its fundamental frequency, then the corresponding wavelength of the string would make it look like this at one moment in time. As the string continue to oscillate, it would then move down to this dashed position and then back to this original one and so forth, on and on. Looking at this string, we can tell that it’s not able to fit an entire wavelength along its length. To do that, to fit an entire wavelength, we would have to go out to a length twice 𝐿. It’s this distance which is equal to what we can call 𝜆, the fundamental wavelength of this string.

Seeing how 𝜆 and 𝐿 relate in the sketch, we can write an equation describing this. The fundamental wavelength 𝜆 of a string that’s fastened at both ends is equal to two times the length of the string 𝐿. This relationship shows us what physical parameters the wavelength 𝜆 depends on. It depends only on the length of the string.

Now, let’s think about the fundamental frequency of this string. We know that when it comes to waves in general, the frequency of a wave — we can call it 𝑓 — is equal to the speed of the wave 𝑣 divided by its wavelength. And we’ve just seen that in this case, 𝜆 is equal to two times 𝐿. So, we can substitute that in for 𝜆 in our equation for 𝑓. It’s worth mentioning that these two relationships for fundamental frequency and fundamental wavelength apply specifically in the case of a string that’s fixed at both ends.

All right, now that we know what parameters the wavelength and the frequency depend on, let’s move on to considering the following descriptions of how frequency and wavelength may change when other string properties change.

Description (a) Frequency: Lower, Wavelength: Longer. (b) Frequency: Lower, Wavelength: Stays the same. (c) Frequency: Stays the same, Wavelength: Shorter. (d) Frequency: Higher, Wavelength: Shorter. (e) Frequency: Higher, Wavelength: Stays the same. When a piano string is replaced with another string of greater density but is placed under the same tension and has the same length, which of the above describes how the fundamental frequency and fundamental wavelength of the string change?

Okay, so, as we saw, in this example, we’re working with a piano string. And we know that when this string vibrates, it has certain properties known as its fundamental frequency and fundamental wavelength. And we saw that if the length of this string is 𝐿, then the fundamental wavelength 𝜆 is equal to two times 𝐿 and the fundamental frequency 𝑓 is equal to the wave speed 𝑣 divided by two 𝐿.

Now, in this description, we’re told that this piano string is replaced with another string of greater density. So, let’s say that this is our new, more dense string. But we’re told that it’s under the same tension and has the same length as the original one. The question is, how does this change to a more dense string affect the fundamental frequency and fundamental wavelength of the string? In other words, how did the fundamental frequency and wavelength of this thicker string compared with this thinner one?

To answer this question, we can rely on these equations which show us what physical parameters 𝑓 and 𝜆 depend on. We see that 𝜆, the fundamental wavelength of a string, only depends on one physical parameter, the string length. Since we’re explicitly told that this more dense string has the same length as the original one, that means the fundamental wavelength of our new string will be the same as that of the old one. And this is because the lengths of these two strings are the same.

Knowing this, we can look over our five answer options and eliminate any which do not claim that the wavelength stays the same. This includes options (a), (c), and (d). These all say that the fundamental wavelength either gets longer or shorter, which we know is not true.

Next, we can consider how this change in string density might affect the fundamental frequency 𝑓. We see that this value depends on two physical parameters, wave speed and string length. We’ve already seen that string length is constant, but what about the characteristic speed of this standing wave? When a wave is moving along a string, its speed — we can call it 𝑣 — is equal to the square root of the force of tension along the string — we’ve called it 𝐹 sub 𝑇 — divided by the string density 𝜇.

Now, in our description, we’re told that this new string does have a greater density but it has the same tension as the original one. So, looking at our equation for wave speed, we see that 𝐹 sub 𝑇 isn’t changing as we moved to the new string but 𝜇 is. It’s going up. Now, if the string density increases, and it’s in the denominator of this equation, then that tells us that increasing 𝜇 will decrease the wave speed 𝑣. And then, looking over at our equation for fundamental frequency, we see that if wave speed 𝑣 goes down, then the fundamental frequency 𝑓 must go down as well. We know this because the string length 𝐿 is constant. And so, if 𝑣 gets smaller, 𝑓 must too.

Therefore, while the fundamental wavelength of this string doesn’t change, its fundamental frequency does. It decreases. This agrees with our answer option (b), which says that the fundamental frequency will get lower while the fundamental wavelength will stay the same. This is what will happen if our string is replaced by another string of greater density but under the same tension and with the same length.

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