A 60.00 cm guitar string under a tension of 46.000 N has a mass per unit length of 0.11000 g/ cm.

(a) What is the speed of the wave? (b) What is the fundamental wavelength for this string? (c) What is the fundamental frequency for this string? (d)What is the highest resonant frequency that can be heard by a person capable of hearing frequencies up to 20,000 Hz?

To answer these questions, we need to use a few formulas related to waves and strings. Let's go step by step:

(a) The speed of the wave (v) on the string can be calculated using the formula:
v = √(T/μ),
where T is the tension in the string and μ is the mass per unit length.

Plugging in the given values:
T = 46.000 N
μ = 0.11000 g/cm = 0.11000 kg/m (converting g/cm to kg/m)

Therefore,
v = √(46.000 N / 0.11000 kg/m)
v = √418181.8 m^2/s^2
v ≈ 646.0 m/s

(b) The fundamental wavelength (λ) for a string can be calculated using the formula:
λ = 2L,
where L is the length of the string.

Here, the length of the string is given as 60.00 cm. Converting it to meters:
L = 60.00 cm = 0.60 m

Therefore,
λ = 2 * 0.60 m
λ = 1.20 m

(c) The fundamental frequency (f) for a string can be calculated using the formula:
f = v / λ,
where v is the speed of the wave and λ is the wavelength.

Plugging in the values we calculated earlier:
f = 646.0 m/s / 1.20 m
f ≈ 538.3 Hz

(d) To determine the highest resonant frequency that can be heard by a person capable of hearing up to 20,000 Hz, we need to consider the harmonic series for the string.

The harmonic series for a string is given by:
fn = nf,
where fn is the frequency of the nth harmonic and f is the fundamental frequency.

We need to find the highest value of n such that nf ≤ 20,000 Hz.

Rearranging the formula, we get:
n = (highest resonant frequency / fundamental frequency)

Plugging in the values:
n = (20,000 Hz / 538.3 Hz)
n ≈ 37.13

Since n must be a whole number, the highest value of n is 37. Therefore, the highest resonant frequency that can be heard is approximately 37 times the fundamental frequency:
highest resonant frequency ≈ (37 * 538.3 Hz)
highest resonant frequency ≈ 19,908 Hz

So, the answers are:
(a) The speed of the wave is approximately 646.0 m/s.
(b) The fundamental wavelength for this string is approximately 1.20 m.
(c) The fundamental frequency for this string is approximately 538.3 Hz.
(d) The highest resonant frequency that can be heard by a person capable of hearing frequencies up to 20,000 Hz is approximately 19,908 Hz.