Abstract
We study the vibration characteristics of an ideal string in the presence of a curved boundary obstacle which is a common constructional feature of sitar, veena and tanpura. It has been well-established that these plucked string instruments have attractive tonal qualities like better harmonicity, presence of amplitude as well as frequency modulations, and appearance of a large number of overtones. The smooth wrapping and unwrapping of the string over the curved obstacle introduces quadratic nonlinearities whose effect on the vibration characteristics is elucidated through a perturbative analysis using the method of multiple scales. Modal interactions facilitated by the obstacle results in neutrally stable mode-locked periodic solutions in our model thereby accounting for the complex frequency and amplitude modulations. In particular, we find a mode-locked state which tends to an equipartition of energy with an increase in the number of modes. Our numerical simulations suggest that a plucked disturbance favors this state and hence, several overtones can be clearly identified in the sound of these plucked instruments. A good qualitative agreement between the results from our idealized model and the sound of these instruments highlights the role of the finite curved bridge in deciding the tonal quality of sitar, veena and tanpura.
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Notes
We observe that sitar and veena also have frets using which a much larger variation in the frequencies can be achieved while the instrument is being played.
We note that there are other fixed points (with at least one of the modal amplitudes as zero which implies that some modes do not participate in the interaction) which have not been reported here. We have verified that these fixed points are unstable (saddles) and hence, are never realized in practical experiments.
Abbreviations
- \(\alpha \) :
-
Constant
- \(\bar{\tau }\), \(\tau \) :
-
Nondimensional time
- \(\bar{x}\) :
-
Nondimensional spatial coordinate
- \(\bar{y}(\bar{x},\, \bar{\tau })\), \(y(x, \tau )\) :
-
Nondimensional transverse displacement of string
- \(\Gamma \) :
-
Wrapped length of the string over the bridge along X
- \(\gamma \) :
-
Nondimensional wrapped length of the string over the bridge
- \(\gamma _n\) :
-
Wrapped length of the string over the bridge at \(\mathcal{O}(\epsilon ^n)\)
- \(\gamma _{\textrm{st}}\) :
-
Static wrapped length of string over bridge
- \(\phi _n\) :
-
Phase angles for nth mode of the string
- \(\rho \) :
-
Density of the string (kg/m)
- \(A_n\) and \(B_n\) :
-
Orthogonal components of amplitude of string for nth mode
- \(A_P\) :
-
Constant
- \(H_r\) :
-
Height of the right boundary of the string (\(X=L\))
- \(R_n\) :
-
Amplitude of string for nth mode in polar coordinates
- \(T_0\) :
-
Original time scale
- \(T_1\) :
-
A slow time scale
- x :
-
Scaled spatial coordinate \(x \in [0,1]\)
- \(Y_B(X)\) :
-
Height of the bridge
- \(y_n(x, T_0, T_1)\) :
-
Transverse displacement of string at \(\mathcal{O}(\epsilon ^n)\)
- \(y_{\textrm{st}}(x)\) :
-
Static transverse displacement of string
- B :
-
Length of the bridge
- b :
-
Nondimensional length of the bridge
- L :
-
Length of the string
- MMS:
-
Method of multiple scales
- ODE:
-
Ordinary differential equation
- PDE:
-
Partial differential equation
- T :
-
Tension in the string (N)
- Y(X, t):
-
Transverse displacement of the string
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Acknowledgements
We thank Anindya Chatterjee, Anurag Gupta and Sovan Lal Das for useful discussions and comments, and the ministry of human resources and development (MHRD), India for financial support as research scholarship to Ashok Mandal. We also thank the anonymous reviewers for their constructive comments.
Funding
Part of the work was funded by ministry of human resources and development (MHRD), India through project MHRD/ME/2013360A.
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Both authors formulated the research plan together. Ashok Mandal executed the plan and worked out the detailed calculations and generated all the results. Both authors interpreted the results and contributed equally to the preparation of the final manuscript.
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Appendices
Appendix A: Discretized set of ODEs from Galerkin projections
For the discretization of our governing PDE (7), we assume the transverse displacement as
Note that this assumed displacement profile trivially satisfies the boundary conditions in (8). Enforcement of the slope continuity boundary condition (9) gives us the wrapped length \(\gamma \) in terms of the modal coordinates \(\beta _n'\)s as
Substituting (38) and (39) in (7) and applying Galerkin projection, we get the discretized set of ordinary differential equations (ODEs) for the modal coordinates (see [3] for details) as
for \(m=1,\, 2,...N\) and the \(K'\)s are defined as
These discretized ODEs are used to numerically study the various properties of the solution so as to verify the findings of the perturbative approach presented in this paper.
Appendix B: PDE at \(\mathcal{O}(\epsilon ^2)\) resulting from (7)
Appendix C: Initial conditions corresponding to plucking at an arbitrary location along the string
An appropriate initial condition for the various modes of the string when plucked at a point corresponds to a triangular shape for the displaced string with the maximum displacement at the point of plucking. For the discretization of our system of equation in “Appendix A”, we have assumed the transverse displacement as given in (38). Multiplying (38) by \(\sin (m \pi x)\) and integrating over the domain \(x \in [0, 1]\) and using the orthogonality properties of the sine series, we get the modal coordinates in terms of the displaced profile \(y(x,\tau )\) as
The initial shape for a string plucked at \(x=x_2\) can be described through two line segments as
where s is the downward displacement of the point \(x=x_2\) measured from the horizontal line \(y=1\) which is the nondimensional height of the bridge. Now the straight line (43) must have the same slope as the slope of the bridge at \(x=0\). Substituting (43) into (9) and simplifying we get
Solving (45) we obtain the wrapped length of the string for the plucked profile as
Now substituting (43), (44) and (46) into (42) we can get the modal coordinates for any possible plucked (triangular) shape of the string. In Table 7 we have listed the initial modal coordinates (for \(N=10\)) for different plucking positions \(x=x_2\) and \(s=x_2+0.1\) which means the string is plucked approximately 0.1 nondimensional length units downward from the static configuration of the string. We note that the slope of the line corresponding to the static configuration of the string is \(\alpha (b-2\gamma _{st}) (1-\gamma _{st})\) with \(\gamma _{st}=1-\sqrt{1-b}\) which is approximately \(-1\) but slightly lower than 1 in magnitude.
Appendix D: Slow flow equations for 3 modes in terms of amplitudes and relative phases
The slow flow equations obtained for the amplitudes and relative phases of a string with 3 modes (\(N=3\)) are
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Mandal, A.K., Wahi, P. Equipartition of modal energy in an ideal string vibrating in the presence of a boundary obstacle. Nonlinear Dyn 112, 215–232 (2024). https://doi.org/10.1007/s11071-023-09046-w
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DOI: https://doi.org/10.1007/s11071-023-09046-w