Radiation force exerted by progressive waves on a string in terms of polarizability

Polarizabilities represent the response of scatterers in the long-wavelength limit. The present work employs polarizabilities to calculate the radiation force exerted on a scatterer of length a and dimensionless mass density μ(x) by 1-D progressive waves on a string. The radiation force density equals 〈∂S/∂x〉according to momentum conservation at quadratic order, where the radiation stress S is obtained in the Born approximation in terms of the polarizabilities α 0 = ∫ a μ(x) dx and α 1 = 2k ∫ a μ(x) x dx, where k is the wavenumber. The radiation force equals 〈S(a/2)–S(–a/2)〉= k 4 ξ 0 2 τα 1 2/8, where ξ 0 is the incident wave amplitude and τ is the string’s tension. The force agrees with solutions based on Fourier transforms for ka « 1 [Morse and Ingard, Theoretical Acoustics (McGraw-Hill, 1968), Eq. (4.5.17)]. To the order of the present approximation, scatterers with μ(x) = μ(–x) do not experience radiation forces due to progressive waves on a string, while scatterers with μ(x) = –μ(–x) experience forces on the order of (ka)6. The results elucidate the approximations underlying the more involved calculation of radiation force exerted by progressive acoustic waves [Gokani et al., JASA 157 (2025); doi: 10.1121/10.0037572].