Does increasing the tension of the string affect the wavelength of the fundamental standing wave on a guitar string?

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To determine whether increasing the tension of a guitar string affects the wavelength of the fundamental standing wave, we can consider the equation that relates the tension (T), linear density (μ), and wavelength (λ) of a wave on a string:

v = √(T/μ)

where v is the wave speed. The wave speed is also given by the formula:

v = fλ

where f is the frequency of the wave. Since the fundamental frequency is the lowest possible frequency, the first harmonic, it means that the wavelength of the fundamental wave (λ1) will be twice the length of the guitar string (L):

λ1 = 2L

Now, let's analyze the impact of changing the tension on the wavelength. Mathematically, if we increase the tension (T), we can rewrite the equation for the fundamental wavelength as:

v = √((T + ΔT)/μ) (where ΔT represents the increased tension)

Since the wave speed (v) is constant for a given string, we can write:

√((T + ΔT)/μ) = f(2L)

Squaring both sides of the equation, we have:

(T + ΔT)/μ = (f(2L))^2

Simplifying further:

(T + ΔT)/μ = 4(f^2)(L^2)

(T + ΔT) = 4(f^2)(L^2)(μ)

Now, let's consider the original equation for the fundamental wavelength (λ1), assuming a constant linear density (μ):

λ1 = 2L

If we compare this with the equation derived after increasing the tension (T + ΔT):

λ'1 = (T + ΔT)/μ

We can see that the increased tension causes a change in the wavelength. So, increasing the tension of the guitar string does affect the wavelength of the fundamental standing wave.

It's important to note that this explanation is based on the assumptions that the linear density remains constant and the string is unchanging in other aspects (e.g., its length and material properties). Real-world instruments might have additional factors that can influence the relationship between tension and wavelength.