A Note on Stefan Wave Problem
Keywords:
finite difference, perturbation solution, Stefan problem, wave equationAbstract
In this paper, the so-called Stefan problem for wave equation is studied. A smooth obstacle placed at one end of a vibrating string causes the effective length of the vibrating string to be changing over time. A straightforward perturbation expansion is used to solve the problem analytically. Some remarks regarding studies of a similar problem by others in the literature are given. Furthermore, we implement numerical method to solve the problem and analyzes the result comprehensively.
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References
N. L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl., 3(3), pp. 472–484, 1961, doi: 10.1016/0022-247X(61)90071-3.
C. Issanchou, J. L. Le Carrou, S. Bilbao, C. Touzé, & O. Doaré, A modal approach to the numerical simulation of a string vibrating against an obstacle: Applications to sound synthesis, DAFx 2016 - Proc. 19th Int. Conf. Digit. Audio Eff., 2, pp. 167–174, 2016.
D. Kartofelev, Kinematics of ideal string vibration against a rigid obstacle,DAFx 2017 - Proc. 20th Int. Conf. Digit. Audio Eff., 1, pp. 40–47, 2017.
B. Pelloni & D. A. Pinotsis, Moving boundary value problems for the wave equation, J. Comput. Appl. Math., 234(6), pp. 1685–1691, 2010, doi: 10.1016/j.cam.2009.08.016.
S. Bilbao, Numerical Modeling of String / Barrier Collisions, Isma2014, no. June, 2014.
C. Issanchou, V. Acary, F. Pérignon, C. Touzé, & J.-L. Le Carrou, Nonsmooth contact dynamics for the numerical simulation of collisions in musical string instruments, J. Acoust. Soc. Am., 143(5), pp. 3195–3205, 2018, doi: 10.1121/1.5039740.
C. P. Vyasarayani, S. Birkett, & J. McPhee, Modeling the dynamics of a vibrating string with a finite distributed unilateral constraint: Application to the sitar, J. Acoust. Soc. Am., 125(6), pp. 3673–3682, 2009, doi: 10.1121/1.3123403.
A. K. Mandal & P. Wahi, Natural frequencies, modeshapes and modal interactions for strings vibrating against an obstacle: Relevance to Sitar and Veena, J. Sound Vib., 338, pp. 42–59, 2015, doi: 10.1016/j.jsv.2014.06.010.
A. K. Mandal & P. Wahi, Mode-locking and improved harmonicity for real strings vibrating in the presence of a curved obstacle, Nonlinear Dyn., 88(3), pp. 2203–2224, 2017, doi: 10.1007/s11071-017-3371-y.
H. Singh & P. Wahi, Role of curvatures in determining the characteristics of a string vibrating against a doubly curved obstacle, J. Sound Vib., 402, pp. 1–13, 2017, doi: 10.1016/j.jsv.2017.04.043.
H. Singh, A. K. Mandal, & P. Wahi, Non-planar motions of a string vibrating against a smooth unilateral obstacle, MATEC Web Conf., 83, pp. 0–3, 2016, doi: 10.1051/matecconf/20168301010
C. Issanchou, J. L. Le Carrou, S. Bilbao, C. Touzé, & O. Doaré, A modal approach to the numerical simulation of a string vibrating against an obstacle: Applications to sound synthesis, DAFx 2016 - Proc. 19th Int. Conf. Digit. Audio Eff., 2, pp. 167–174, 2016.
D. Kartofelev, Kinematics of ideal string vibration against a rigid obstacle,DAFx 2017 - Proc. 20th Int. Conf. Digit. Audio Eff., 1, pp. 40–47, 2017.
B. Pelloni & D. A. Pinotsis, Moving boundary value problems for the wave equation, J. Comput. Appl. Math., 234(6), pp. 1685–1691, 2010, doi: 10.1016/j.cam.2009.08.016.
S. Bilbao, Numerical Modeling of String / Barrier Collisions, Isma2014, no. June, 2014.
C. Issanchou, V. Acary, F. Pérignon, C. Touzé, & J.-L. Le Carrou, Nonsmooth contact dynamics for the numerical simulation of collisions in musical string instruments, J. Acoust. Soc. Am., 143(5), pp. 3195–3205, 2018, doi: 10.1121/1.5039740.
C. P. Vyasarayani, S. Birkett, & J. McPhee, Modeling the dynamics of a vibrating string with a finite distributed unilateral constraint: Application to the sitar, J. Acoust. Soc. Am., 125(6), pp. 3673–3682, 2009, doi: 10.1121/1.3123403.
A. K. Mandal & P. Wahi, Natural frequencies, modeshapes and modal interactions for strings vibrating against an obstacle: Relevance to Sitar and Veena, J. Sound Vib., 338, pp. 42–59, 2015, doi: 10.1016/j.jsv.2014.06.010.
A. K. Mandal & P. Wahi, Mode-locking and improved harmonicity for real strings vibrating in the presence of a curved obstacle, Nonlinear Dyn., 88(3), pp. 2203–2224, 2017, doi: 10.1007/s11071-017-3371-y.
H. Singh & P. Wahi, Role of curvatures in determining the characteristics of a string vibrating against a doubly curved obstacle, J. Sound Vib., 402, pp. 1–13, 2017, doi: 10.1016/j.jsv.2017.04.043.
H. Singh, A. K. Mandal, & P. Wahi, Non-planar motions of a string vibrating against a smooth unilateral obstacle, MATEC Web Conf., 83, pp. 0–3, 2016, doi: 10.1051/matecconf/20168301010
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Published
2022-10-12
How to Cite
Ihsan, A. F., & Tuwankotta, J. M. (2022). A Note on Stefan Wave Problem. ITB Graduate School Conference, 1(1), 197–207. Retrieved from https://gcs.itb.ac.id/proceeding-igsc/index.php/igsc/article/view/19
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