a) Obtain an expression for energy transported by a progressive wave on a taut string. Also show that the power transported by the wave is proportional to the wave velocity.
Answer:
The energy transported by a traveling wave on a taut string can be analyzed by considering the kinetic and potential energies of a small element of the string. Let’s derive the expressions for these energies and then calculate the total energy and power transported by the wave.
Kinetic Energy:
For a small element Delta x\Delta x of the string, the kinetic energy (d(K.E.)d(K.E.)) is given by:
The power (PP) of the wave is the energy averaged over one time period (TT) and is given by:
P=(E)/(T)=(E)/((lambda//v))=(1)/(2)rhoa^(2)omega^(2)v=2pi^(2)rhoa^(2)f^(2)vP = \frac{E}{T} = \frac{E}{(\lambda / v)} = \frac{1}{2} \rho a^2 \omega^2 v = 2 \pi^2 \rho a^2 f^2 v
where omega=2pi f\omega = 2 \pi f and ff is the frequency of the wave.
Conclusion:
The energy transported by a traveling wave on a taut string is distributed between kinetic and potential energies, with the total energy contained in one wavelength given by (1)/(2)rhoa^(2)omega^(2)lambda\frac{1}{2} \rho a^2 \omega^2 \lambda. The power transported by the wave is proportional to the wave velocity, as well as the square of its amplitude and frequency.
b) The fundamental frequency of a string instrument is 580Hz580 \mathrm{~Hz}. (i) Calculate the frequency of the first and third harmonics generated on it. (ii) If the tension in the string is doubled, calculate the new fundamental frequency.
Answer:
To address this problem, we’ll break it down into two parts as requested:
Part (i): Frequency of the First and Third Harmonics
The fundamental frequency of a string instrument is given as f_(1)=580Hzf_1 = 580 \, \mathrm{Hz}. The harmonics of a string instrument are integer multiples of the fundamental frequency. Thus, the frequency of the nnth harmonic can be calculated using the formula:
f_(n)=n*f_(1)f_n = n \cdot f_1
For the first harmonic (n=1n=1), this is simply the fundamental frequency itself, f_(1)=580Hzf_1 = 580 \, \mathrm{Hz}.
For the third harmonic (n=3n=3), the frequency is f_(3)=3*f_(1)f_3 = 3 \cdot f_1.
Let’s substitute the values to find f_(3)f_3:
f_(3)=3*580Hzf_3 = 3 \cdot 580 \, \mathrm{Hz}
After Calculating, we get:
f_(3)=1740Hzf_3 = 1740 \, \mathrm{Hz}
Part (ii): New Fundamental Frequency if the Tension is Doubled
The fundamental frequency of a string can also be expressed in terms of the tension (TT) in the string, its length (LL), and its mass per unit length (mu\mu) as follows:
When the tension in the string is doubled (T^(‘)=2TT’ = 2T), the new fundamental frequency (f_(1)^(‘)f_1′) can be calculated using the modified tension: