Hyperbolic cosine

Shibuya (SlO) dealt with the case of the onset of instability in film flow on the outer surface of a vertical tube. By assuming a mixed disturbing velocity of the cosine-hyperbolic cosine type, it was found that the numerical value of the Reynolds number for instability was approximately... 

Here, the sine and cosine hyperbolic functions (Sh and Ch) are well-known expressions. The average value of the residence time in the way 0-H can be described with the assistance of the characteristic function ... 

Equation 3-56 is a linear, homogeneous, second-order differential equation with constant coefficients. A fundamental theory of differential equations states (hat such an equation has two linearly independent solution functions, and its general solution is the linear combination of those two solution functions. A careful examination of the differential equation reveals that subtracting a constant multiple of the solution function 0 from its second derivative yields zero. Thus we conclude that the function 0 and its second derivative must be constant multiples of each other. The only functions whose derivatives are constant multiples of the functions themselves are the exponential functions (or a linear combination of exponential functions such as sine and cosine hyperbolic functions). Therefore, the solution functions of the differential equation above are the exponential functions e or or constant multiples of them. This can be verified by direct substitution. For example, the second derivative of e is and its substitution into Eq. 3-56... 

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

Hyperbolic Functions sinh z = (e2 — e )/2 cosh z = e + 2)/2 tanh z = sinh z/cosh z coth z = cosh z/sinh z csch z = 1/sinh z sech z = 1/coshz. Identities are cosh2z - sinh2z = 1 sinh (z, + z2) = sinh z, cosh z2 + cosh Zi sinh z2 cosh (z, + z2) = cosh Zi cosh z2 + sinh Zi sinh z2 cosh z + sinh z =et cosh z - sinh z = e. The hyperbolic sine and hyperbolic cosine are periodic functions with the imaginary period 27ti. That is, sinh (z + 2iti) = sinh z. 

The Thiele modulus is related to the concentration dependence in a catalyst body by the following equations representing the ratios of the hyperbolic cosines ... 

One expects that the differential capacity of a semiconductor/electrnlyte interface due to the space charge inside an intrinsic semiconductor will vary in a hyperbolic cosine manner with the potential. Such a variation is shown in Fig. 6.127. 

The hyperbolic sine and cosine functions sinh - and cosh. v are defined in terms of the sum and difference of the exponential functions e and e. respectively ... 

The coefficients in the above series in if/p(r) alternate between the hyperbolic sine and cosine value of the uniform contribution, i/r0(z). In contrast to a full linear treatment, which is the usual procedure followed, the 0(if/p) term here does not vanish. As if/0(z) is large we must regard it as satisfying the nonlinear, ordinary differential form of the PB equation,... 

An even more simple result is obtained if we use the asymptotic expression for the hyperbolic cosine. In this case,... 

The integrals I0 and /2 arise from a Fourier expansion for hyperbolic cosines of trigonometric argument [45]. With this potential, Parker calculated both the average number of collisions to establish rotational equilibrium, Zf, and that required to establish vibrational equilibrium, Z , as functions of temperature. One finds... 

Under the assumptions employed, the parabolic shape of electrocapillary curves in this model is expressed as the hyperbolic cosine of Aq Electrocapillary curves calculated as 25°C and z = 1 using Equation (15) are shown for ec/so = 8 and 80, where So is the vacuum permittivity, as dashed lines in Figures 7.2a and 7.2b, respectively. It is seen that the curvature of the electrocapillary curves becomes greater with increasing value of sc. 

Fig. 4 (a) Schematic of MIM stracture with gap d. Transverse magnetic field is shown in red, which has a hyperbolic cosine dependence in the gap and exponential decay into the metal, (b) Effective refractive index of waveguide mode within MIM gap, which increases as the gap is made smaller. Calculations for two free-space wavelengths are shown using the relative permittivity of gold and the difference is used to estimate the group index ( g), which also increases as the gap is made smaller. Therefore, the light slows down as the gap is made smaller... 

Figure 4(a) shows a metal-insulator-metal (MIM) gap. The solution to the gap is equivalent to that for a TM dielectric waveguide, except that the relative permittivity of the cladding layers is negative and the field is a hyperbolic cosine inside the gap. The equation to find the solution for the effective index of the MIM waveguide mode is as follows ... 

If we carefully observe the expression of the characteristic function of the residence time distribution for the evolution of a liquid element (q>(s,Hd), relation (4.201)), we can notice that it is difficult to compute the expressions of the derivatives cp (0, Hd) and (p"(0, H ). Using the expansion of the hyperbolic sine and cosine respectively as multiplication series, we obtain the following simplified expression for the characteristic function ... 

This function Is plotted in fig. 3.6. Because of the hyperbolic cosine dependency and because both 2U"e proportional to there is some similarity with the differential capacitance, fig. 3.5. The physical background Is that both quantities depend In a similar way to screening. Writing AG as... 

ACOSH Returns the inverse hyperbolic cosine of a number. 

If the time step is not small (such that At" > 2M/an initial value solver becomes unstable. A solution to the finite difference equation can be found in which A is imaginary, and the cosine function is replaced by a hyperbolic cosine. This solution is not only a poor approximation but it is also numerically unstable. Using initial value solvers the above solution leads to exponential (unbound) growth of the coordinate as a function of time = Aexp(A(i)j)). 

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