Final answer:
If the mass density of a string is doubled while holding the frequency constant, the wavenumber of a traveling wave on the string will be halved.
Step-by-step explanation:
The wavenumber of a traveling wave on a string is given by the equation k = 2π/λ, where k is the wavenumber and λ is the wavelength. The wave speed on the string depends on the tension and the linear mass density, according to the equation v = √(T/μ), where v is the wave speed, T is the tension, and μ is the linear mass density.
If we hold the frequency constant and double the mass density, we can assume that the linear mass density (μ) has doubled. From the equation v = √(T/μ), we can see that as μ increases, the wave speed decreases. Since the frequency (f) is constant and the wave speed (v) is inversely proportional to the wavelength (λ), we can conclude that when the mass density is doubled, the wavelength must also double.
To find the wavenumber, we can use the equation k = 2π/λ. If the wavelength doubles, the wavenumber must halve. Therefore, if the mass density is doubled and the frequency is held constant, the wavenumber of the traveling wave on the string will be halved.